Improved LP-based Approximation Algorithms for Facility Location with Hard Capacities
% We present LP-based approximation algorithms for the capacitated facility location problem (CFL), a long-standing problem with intriguing unsettled approximability in the literature dated back to the 90s. % We present an elegant iterative rounding scheme for the MFN relaxation that yields an approximation guarantee of $\left(10+\sqrt{67\right)/2 \approx 9.0927$, a significant improvement upon the previous LP-based ratio of $288$ due to An et al. in~2014. % \smallskip % For CFL with cardinality facility cost (CFL-CFC), %i.e., assuming the cardinality of facilities instead of weighted cost, we present an LP-based $4$-approximation algorithm, which not only surpasses the long-standing ratio of~$5$ due to Levi et al. that ages up for decades since 2004 but also unties the long-time match to the best approximation for CFL that is obtained via local search in 2012. % Our result considerably deepens the current understanding for the CFL problem and indicates that an LP-based ratio strictly better than $5$ in polynomial time for the general problem may still be possible to pursue. %
Final-Version-Improved_approx_CFL-general-FOCS.tex
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\title{\bf Improved LP-based Approximation Algorithms for Facility Location with Hard Capacities}

\author[ ]{Mong-Jen Kao}

\affil[ ]{Department of Computer Science and Information Engineering,}
\affil[ ]{National Chung-Cheng University, Taiwan.}
\affil[ ]{\small Email: \texttt{mjkao@alum.ccu.edu.tw}}


\date{}
%\date{\small\today}

\pagestyle{plain}

\begin{document}

%\begin{titlepage}

\maketitle
%\thispagestyle{empty}

\begin{abstract}
%
We present LP-based 
approximation 
algorithms for the capacitated facility location problem (CFL), a long-standing problem with intriguing unsettled approximability in the literature dated back to the 90s.
%
We present an elegant iterative rounding scheme for the MFN relaxation that yields an approximation guarantee of $\left(10+\sqrt{67}\right)/2 \approx 9.0927$, a significant improvement upon the previous LP-based ratio of $288$ due to An et al. in~2014.
%

\smallskip

%
For CFL with cardinality facility cost (CFL-CFC), 
%i.e., assuming the cardinality of facilities instead of weighted cost, 
we present an LP-based $4$-approximation algorithm,
which not only surpasses the long-standing ratio of~$5$ due to Levi et al. that ages up 
for decades since 2004 
but also unties the long-time match to the best approximation for CFL that is obtained via local search in 2012.
%
Our result considerably deepens the current understanding for the CFL problem and indicates that an LP-based ratio strictly better than $5$ in polynomial time 
for the general problem 
may still be possible to pursue.
%
\end{abstract}

%\newpage

%\bigskip

%\begin{minipage}{0.95\textwidth}
%\textbf{Keywords:} iterative partial rounding, capacitated vertex cover, hard capacities, approximation algorithm
%\end{minipage}

%
%\end{titlepage}

%

\section{Introduction}

%
We consider the facility location problem with hard capacities (CFL), a long-standing problem with intriguing unsettled approximability and literature dated back to the 90s.
%
In this problem, we are given a set $\MC{F}$ of facilities, a set $\MC{D}$ of clients, and a distance metric $c$ defined over $\MC{F} \cup \MC{D}$.
%
Each $i \in \MC{F}$ is associated with an open cost $o_i$ and a capacity $u_i$,
%, 
which is the number of clients it can serve when opened up.
%
The cost of serving a client $j$ using a facility $i$ equals the distance 
%$c_{i,j}$ 
between them.
%
The goal is to determine a set of facilities to open up and an assignment of the clients to the opened facilities which respects their capacity 
%constraints 
limits so that the total cost is minimized.
% for opening up the facilities plus the cost for serving the clients is minimized.
%

\smallskip

The CFL problem
%facility location problem with hard capacities, or, capacitated facility location problem (CFL), is a long-standing problem with unsettled approximability since it 
was first considered by Shmoys, Tardos, and Aardal~\cite{10.1145/258533.258600}. 
%Since then, almost all results for this problem were based on local search heuristics. 
%\mong{uniform capacity, soft capacity *** double check these results ***}
%
For facilities with uniform capacities, Koropolu et al.~\cite{10.5555/314613.314616} showed that the local search heuristic proposed by Kuehn and Hamburger~\cite{10.1287/mnsc.9.4.643} yields a constant factor approximation.
%
Chudak and Williamson~\cite{10.5555/645589.659788} improved the analysis of Korupolu et al.~\cite{10.5555/314613.314616} and obtained a $(6,5)$-approximation, i.e., a solution whose cost is bounded by 6 times the facility cost plus 5 times the service cost of an optimal solution.
%
Aggarwal et al.~\cite{DBLP:journals/mp/AggarwalLBGGGJ13} introduced the idea of taking suitable linear combinations of inequalities which captures the local optimality and obtained a $3$-approximation.
%

\smallskip

%
For facilities with non-uniform capacities, i.e., the general CFL problem, Pal et al.~\cite{10.5555/874063.875600} presented a $(9,5)$-approximation based on local search algorithm.
% was due to .
%
The ratio was improved by Mahdian and Pal~\cite{DBLP:conf/esa/MahdianP03} to an $(8,7)$-approximation.
%
Zhang et al.~\cite{DBLP:conf/ipco/ZhangCY04} introduced the idea of multi-exchange local operations and further improved the ratio to $(6,5)$.
% by introducing a .
%
The algorithm was later modified by Bansal et al.~\cite{DBLP:conf/esa/BansalGG12} to achieve a $5$-approximation, which is the best ratio known for the CFL problem.
%

\smallskip

%
In contrast to the rich LP-based toolboxes developed for the uncapacitated facility location problem (UFL), the fact that no LP-based algorithms with constant approximation guarantee were known for CFL was intriguing and surprising.
%
In fact, devising an LP-based approximation with constant guarantee for CFL was listed as one of ten open problems in the textbook on approximation algorithm due to Williamson and Shmoys~\cite{10.5555/1971947}.
%
This problem was resolved by the notable work
%partially 
of An, Singh, and Svensson~\cite{DBLP:conf/focs/AnSS14}, in which a strong multi-commodity flow network (MFN)
%LP 
relaxation
%, the first LP relaxation 
with constant integrality gap is presented.
%
The approximation guarantee obtained in this work, however, is in the order of couple hundreds, and it remains an open problem to devise a better LP-based guarantee for CFL or a better integrality gap for the MFN relaxation.
% \mong{*** approximation algorithm with guarantee in the order of hundreds ***}
%

\medskip

%
In the pursuit of settling down the approximability of CFL, an important variation between the general problem and the case with uniform capacities is when we have 
%(non-uniform capacities and 
cardinality-type facility costs, i.e., uniform facility cost,
% where $o_i = 1$ for all $i \in \MC{F}$, 
%i.e., the cardinality-type facility cost.
%
This was studied by Levi, Shmoys, and Swamy~\cite{DBLP:conf/ipco/LeviSS04}, in which an LP-based $5$-approximation 
%for this problem 
was presented.
%
Interestingly, the approximation ratio of $5$ 
%for the case with uniform facility cost 
not only remains the best known ratio for a long time since 2004, but also coincide with the best known approximation result obtained later for the general CFL problem~\cite{DBLP:conf/esa/BansalGG12} using local search heuristic.

%

%\paragraph{Related work.}

%
%When multiple instances of the facilities are allowed, this problem is known as soft capacitated facility location problem (SCFL).
%
%Chudak and Shmoys~\cite{10.5555/314500.315061} gave a $3$-approximation for SCFL with uniform capacities using LP-rounding.
%
%For non-uniform capacities, Jain and Vazirani~\cite{10.1145/375827.375845} reduced the SCFL problem to uncapacitated facility location (UFL) and obtained a $4$-approximation via primal-dual method.
%
%Arya~\cite{10.1145/380752.380755} proposed a local search algorithm and obtained an approximation ratio of $3.72$.
%
%Jain, Mahdian, and Saberi~\cite{10.1145/509907.510012} obtained a $3$-approximation following the approach of Jain and Vazirani~\cite{10.1145/375827.375845}.
%
%The ratio was improved by Mahdian et al.~\cite{10.5555/646689.703120} to $2.89$ and further improved to $2$ by the same authors~\cite{DBLP:conf/random/MahdianYZ03}.
%

%\smallskip

%
%\mong{*** capacitated problems ***}

%

%

\subsection{Our Contributions}
%\paragraph{Our Results.}

We present improved LP-based approximation algorithms for the general CFL problem and the case with cardinality facility costs.
%
Our first result is the following theorem.
%

\begin{theorem} \label{thm-approx-cfl}
There is an LP-based algorithm for CFL that produces a $\left(10+\sqrt{67}\right)/2 \approx 9.0927$-approximation in polynomial-time.
\end{theorem}

%
This yields a significant improvement upon the previous LP-based ratio of 288 due to An~et~al.~\cite{DBLP:conf/focs/AnSS14} in~2014.
%
Our approximation algorithm for CFL is built on an elegant iterative rounding scheme for the MFN relaxation that combines the ideas from several previous works~\cite{zakmsp02,DBLP:conf/ipco/LeviSS04,DBLP:conf/focs/AnSS14}.
%
%Although our algorithm inherits the framework from~\cite{DBLP:conf/focs/AnSS14}, the details as well as the philosophies behind are quite distinguishable.
%
We believe that our result considerably deepens the current understanding for the CFL problem.
%

%\smallskip

%
%Our approach inherits the ``round-or-separate`` framework from An~et~al.~\cite{DBLP:conf/focs/AnSS14}. 
%
%The details and the ideas, however, are quite distinguishable.
%

%
%\mong{**** the details are quite distinguishable ****}
%

\smallskip

%
The second result we present is the following theorem for capacitated facility location problem with cardinality facility cost (CFL-CFC).
%, for which we denote by CFL-CFC.
%\mong{*** CFL-UFC ***}

\begin{theorem} \label{thm-approx-cfl-ufc}
There is an LP-based algorithm for 
%the capacitated facility location with uniform facility cost
CFL-CFC that produces a $4$-approximation in polynomial-time.
\end{theorem}

%Our result
This surpasses the decades-old ratio of $5$ due to Levi et al.~\cite{DBLP:conf/esa/BansalGG12} which ages up 
since 2004 and unties the long-time match to the best approximation for CFL that is obtained via local search in 2012.
%
This sends a message that, an LP-based ratio strictly better than $5$ for the general CFL problem may still be possible to pursue.
%\cite{DBLP:conf/ipco/LeviSS04}
%
%The technical tool for establishing this result 
%
%\smallskip
%
%
Our algorithm for CFL-CFC is built on a two-staged rounding scheme that combines a novel set of ideas with 
%the ideas 
%from several prior works~\cite{10.5555/3039686.3039860,DBLP:conf/ipco/LeviSS04,DBLP:conf/focs/AnSS14,zakmsp02} 
the results developed in the past for both facility location and capacitated covering problems~\cite{10.5555/3039686.3039860,DBLP:conf/soda/CheungGW14,zakmsp02,DBLP:conf/ipco/LeviSS04}. 
%

%\smallskip

%
%\mong{****}
%The result we present
%Our results considerably deepen the current understanding for the CFL problem 
%and indicate that an LP-based ratio strictly better than $5$ for the general problem may still be possible to pursue.

%

%


%%
%%

\subsubsection*{Overview of the Algorithms}
%\paragraph{Overview of the Algorithms.}

The core part of our results can be seen as rounding procedures for small instances incurred in the LP relaxations,
% and the natural LP relaxation, respectively, 
i.e., the rounding decisions for the small facilities along with the assignments made to them.
%
Our proposed procedures aim at fractionally serving the clients while making sure that the rounded facilities of interests (if not all) are reasonably sparsely-loaded by the assignments.
%Depending on the underlying working LP relaxation, the actual algorithm design appears differently.
%

\smallskip

%
Our algorithm for CFL inherits the ``round-or-separate`` framework from An~et~al.~\cite{DBLP:conf/focs/AnSS14} with a distinguishable set of ideas and philosophies behind.
%
Inside each iteration incurred in this framework, we are given as input a candidate solution and the task is either to round this candidate 
solution integrally, or to produce a 
%proof invalidating the candidate solution
separating hyperplane so that 
%the 
Ellipsoid method can proceed.
%
Our result for this part is the following theorem.

\begin{theorem*}[Sketch]
%\label{thm-MFN-relaxed-separation-oracle}
%
Given 
%target parameter $\gamma$ and 
a candidate solution $(\bm{x}, \bm{y})$ for the MFN relaxation and a target parameter $\alpha$ with $0 < \alpha \le 1/3$, we can compute in polynomial-time either (i) a 
%constraint violated by
hyperplane separating $(\bm{x}, \bm{y})$ 
%for the relaxation,
and the feasible region, or (ii) an integral solution $(\bm{x}^*, \bm{y}^*)$ rounded from $(\bm{x}, \bm{y})$
such that
%for $\Psi$ with 
%$\psi(\bm{x}^*, \bm{y}^*) \le ??\cdot \gamma$. \mong{*** pin down the ratio ***}
%
$$\text{cost}(x^*, y^*) \le \max\left\{ \; \frac{3}{2\alpha} \; , \; \frac{7-4\alpha}{(1-\alpha)^2} \; \right\} \times \text{cost}(x,y).$$
\end{theorem*}

%
In our algorithm, instead of guaranteeing a firm fraction of demand be sent to the small instances, as was did in the previous work by An et al.~\cite{DBLP:conf/focs/AnSS14}, we emphasize that, with proper construction, the large facilities are sparsely-loaded by the flow sent to them and therefore are ready for a reasonable rounding blow-up of $1/(1-\alpha)$ when necessary.
%

\smallskip

%
Our rounding procedure for the small facilities is built on the idea due to Abraham et al.~\cite{zakmsp02} and a couple of prior works developed for uncapacitated facility location.
%
In each iteration, the facility with the least ``per-unit-flow-rerouting-cost``, i.e., the cost incurred if we relocate flow \emph{simultaneously} and \emph{proportionally} from \emph{all facilities in the vicinity} to the facility, is selected to be rounded.
%
%Intuitively, the ``per-unit-flow-rerouting-csot`` of facility $i$ is the cost incurred if we relocate flow \emph{simultaneously} and \emph{proportionally} from \emph{all facilities in the vicinity} of $i$ to $i$.
%
%\smallskip
%
%
%From another perspective, we are to scale up both the facility value of $i$ and all the assignments made to $i$ at the same time, and distribute the cost incurred to the demand (flow) we gathered from the vicinity.
%
%Since $$\alpha\cdot \frac{1}{y_i} \sum_{j} x_{i,j} \; \le \; \alpha \cdot \sum_{j} r_j$$ holds by the LP feasibility, provided that $\sum_{i} x_{i,j} \ge \alpha$ for all $j$, we know that there will be an adequate amount of demand in the vicinity to be gathered.
%
After the rounding process is done, we guarantees that at least $(1-\alpha)$-fraction of each client is assigned.
%

\smallskip

%
To bound the extra rerouting cost incurred during the process, in the previous works~\cite{DBLP:conf/focs/AnSS14,zakmsp02} a filtering technique is applied to ensure a low assignment radius for each client throughout the process.
%
However, this technique inevitably causes a tremendous blowup in the resulting guarantee.
%
In our result, we apply an implicit primal-dual schema for an exact pricing on the cost incurred.
%
This 
%, which in turn 
bounds the assignment radius tightly for each client that gets reassigned.
%
%We reevaluate the primal-dual schema 
To ensure that correct pricing 
%for the remaining instance 
is obtained for the need of our rounding process in each iteration, we apply this technique in an iterative manner.
%rounding technique.
%
Together this yields our approximation for CFL.

\medskip

%
Our approximation algorithm for CFL-CFC is built on a two-staged rounding scheme that combines a novel set of ideas with the results developed in the past for both facility location and covering problems~\cite{10.5555/3039686.3039860,DBLP:conf/soda/CheungGW14,zakmsp02,DBLP:conf/ipco/LeviSS04}. 
%
As is done in our first result, we apply primal-dual schema to ensure a tightly-bounded assignment radius for each client throughout the rounding process.
%
However, the large facilities are no longer sparsely-loaded by the assignment given by natural LPs.
%

\smallskip

%
To overcome this issue, when the residue demand of a client drops below the target threshold, we discard the residue client and redistribute part of it to the large facilities in the vicinity to form the so-called ``\emph{outlier clients}.``
%
The outlier clients participate in the rounding process and act as normal clients except for that, there is no threshold for the outlier clients to be discarded, and we guarantee that the outlier clients will be fully-assigned for the final feasibility.
%

\smallskip

%
To achieve this guarantee, in the second stage of the rounding process, we formulate the assignment problem of the remaining outlier clients as a 
carefully designed 
%simple 
assignment LP.
%
We apply a technique, which is originally developed for the capacitated covering problems~\cite{DBLP:conf/soda/CheungGW14,10.5555/3039686.3039860}, to show that, basic feasible solutions of this simple LP corresponds naturally to an implicit matching between the large facilities at which the outlier clients reside and the non-integral facilities given by this LP.
%
Together this yields a bound for our final rounding.
%

%

%%%%
%%%%

\paragraph{Organization of this paper.}

The rest of this paper is organized as follows.
%
In the rest of this section we describe the preliminaries necessary to present our approximation algorithms.
%
We present our approximation algorithm for CFL in Section~\ref{sec-approx-cfl}, page~\pageref{sec-approx-cfl}, and our approximation algorithm for CFL-CFC in Section~\ref{sec-approx-cfl-cfc}, page~\pageref{sec-approx-cfl-cfc}.
%
%
%\smallskip
%
%
%In the interests of a better reading experience, we included the algorithmic ideas, LP formulations, and math equations with ample space in the main content, 
%which slightly exceeds the 10-page limit. \mong{****** no longer exceeds the page limits ******}
%

\smallskip

%
The additional content is organized as follows.
%
We establish our approximation guarantee for CFL in Section~\ref{sec-proof-approx-cfl}, page~\pageref{sec-proof-approx-cfl}, and our guarantee for CFL-CFC in Section~\ref{sec-proof-approx-cfl-cfc}, page~\pageref{sec-proof-approx-cfl-cfc}.
%

%

%

%%
%%

\subsection{Preliminaries}

%

%\mong{**** $\Psi = (F,D,c,o,m,r)$ ***}
%
%\mong{**** CFL ****}
%
In the CFL problem, we are given a set $\MC{F}$ of facilities, a set $\MC{D}$ of clients, and a distance metric $c$ defined over $\MC{F} \cup \MC{D}$.
%
Each $i \in \MC{F}$ is associated with an open cost $o_i$ and a capacity $u_i$, which is the number of clients it can serve when opened up.
%
%
%\smallskip
%
%
%
A facility multiplicity function $y\colon \MC{F} \rightarrow [0,1]$
%, which 
%. Intuitively, $y_i$ 
denotes the decision whether each facility 
%$i$ 
is selected to be opened up.
%
A client assignment 
%is 
%defined to be 
%a function 
$x\colon \MC{F} \times \MC{D} \rightarrow [0,1]$
%, where $x_{i,j}$ denotes 
specifies the decision whether client $j$ is assigned to facility $i$ (served by facility $i$).
%
%
%\smallskip
%
A solution for CFL consists of a multiplicity function $y$ and an assignment function $x$ such that the following conditions are met:
%
%\begin{itemize}
%	\item
		(a) $\sum_{i \in \MC{F}}x_{i,j} \ge 1$, for each $j \in \MC{D}$. %That is, each client is assigned to some facility.
%	
%	\item
		(b) $\sum_{j \in \MC{D}}x_{i,j} \le u_i \cdot y_i$, for each $i \in \MC{F}$. %That is, the number of clients assigned to any facility does not exceed its available capacity.
%		
%	\item
		(c) $x_{i,j} \le y_i$, for each $i \in \MC{F}, j \in \MC{D}$.
%		
%\end{itemize}
%
%\smallskip
%
%\noindent
The cost of a solution $(x,y)$ is defined to be 
$$\psi(x,y) \; := \; \sum_{i \in \MC{F}} o_i \cdot y_i \; + \; \sum_{i \in \MC{F}, \; j \in \MC{D}} c_{i,j} \cdot x_{i,j}.$$
%
Given an instance $\Psi = (\MC{F},\MC{D}, \MB{c}, \MB{o}, \MB{u})$ of CFL,
the goal of this problem is to compute an integral solution $(x,y)$ such that $\psi(x,y)$ 
%the sum of facility cost, $\sum_{i \in \MC{F}}y_i$, and assignment cost, $\sum_{i \in \MC{F}}\sum_{j \in \MC{D}} c_{i,j}\cdot x_{i,j}$, 
is minimized.

%

%

%\smallskip


%%%
%%%

%\paragraph{Natural LP Relaxation.}
%A natural LP relaxation for the CFL problem is formulated as follows.

%

%\begin{figure*}[h]
%\centering
%\fbox{
%\begin{minipage}{.8\textwidth}
%\begin{align}
%& \text{min} & & \sum_{i \in \MC{F}} o_i \cdot y_i \; + \; \sum_{i \in \MC{F}, \hspace{2pt} j \in \MC{D}} c_{i,j}\cdot x_{i,j} & &  \label{LP-natural-CFL} \tag*{LP-(N)} \\[3pt]
%& \text{s.t.} & & \sum_{i\in \MC{F}}x_{i,j} \; \ge \; 1, & & \forall j\in \MC{D} \tag*{(P-1)} \\[3pt]
%& & & \sum_{j \in \MC{D}} x_{i,j} \; \le \; u_i \cdot y_i, & & \forall i \in \MC{F} \label{cons-P-2} \tag*{(P-2)} \\[3pt]
%& & & 0 \; \le \; x_{i,j} \; \le \; y_i, & & \forall i \in \MC{F}, j \in \MC{D} \quad \label{cons-P-3} \tag*{(P-3)} \\[5pt]
%& & & 0 \; \le \; y_i \; \le \; 1, & & \forall i \in \MC{F}. \label{cons-P-4} \tag*{(P-4)}
%\end{align}
%\vspace{-6pt}
%\end{minipage}\quad
%}
%\caption{A natural LP relaxation for CFL.}
%\label{fig-natural-LP}
%\end{figure*}

%

%%%
%%%

\paragraph{The MFN Relaxation.}

%The simple natural LP does not have bounded integrality gap in general.
%
%This is illustrated by \mong{**** simple example **** blahblah}
As the natural LP formulation for CFL is known to have an unbounded integrality gap even for simple settings,
%
%\smallskip
%
%
An, Singh, and Svensson~\cite{DBLP:conf/focs/AnSS14} designed a strong LP relaxation based on multicommodity flow networks (MFN).
% for this problem.
%
The idea is to impose Knapsack-cover type constraints, formulated as reassignable \emph{partial assignments} given as free in each qualifying test on both the facility values and the assignments of the clients.
%
In the following we introduce the framework and the construction of this relaxation.
%

\begin{definition}[Partial Assignments]
%
A partial assignment $g$ is 
%an assignment
a function $g\colon \MC{F} \times \MC{D} \rightarrow [0,1]$.
%
The partial assignment $g$ is said to be valid if
%\begin{itemize}
%	\item
		\; (i) $\hspace{1pt} \sum_{i \in \MC{F}} g_{i,j} \le 1$, for each $j \in \MC{D}$, and
%		
%	\item
		\; (ii) $\hspace{1pt} \sum_{j \in \MC{D}} g_{i,j} \le u_i$, for each $i \in \MC{F}$.
%\end{itemize}
%
%\mong{*** occupying *** if $g_{i,j} > 0$ implies that $\sum_{j \in \MC{D}} g_{i,j} = u_i$. ****}
%
%A partial assignment $g$ is additionally referred to as $(\alpha,y)$-concentrated for some $0 < \alpha < 1$ and multiplicity function $y$ if for all $i\in \MC{F}, j \in \MC{D}$, $g_{i,j} > 0$ implies that $y_i \ge \alpha$.
\end{definition}

%

\smallskip


Given a candidate fractional solution $(x,y)$ and a valid partial assignment $g$, An, Singh, and Svensson~\cite{DBLP:conf/focs/AnSS14} 
%proposed 
designed the following multi-commodity flow network, denoted $\mathbf{MFN}_\Psi(x,y,g)$, as a qualifying test for the candidate solution $(x,y)$.
%

\begin{definition}[Multi-commodity Flow Network]
%
For a valid partial assignment $g$ and a candidate solution $(x,y)$, $\mathbf{MFN}_\Psi(x,y,g)$ is a multicommodity flow network 
%with $\left|\MC{D}\right|$ commodities, 
defined as follows.
\begin{itemize}
	\item
		Each client $j \in \MC{D}$ corresponds to two nodes $j^s$ and $j^t$ in the network and is associated with a commodity $j$ with source-sink pair $(j^s, j^t)$ and demand $r^{(g)}_j := 1-\sum_{i \in \MC{F}} g_{i,j}$.
		
	\item
		Each facility $i \in \MC{F}$ corresponds to two nodes $i$ and $i^t$ that are connected by an arc $(i,i^t)$ of capacity $u^{(g)}_i := y_i \cdot \left( u_i - \sum_{j \in \MC{D}} g_{i,j} \right)$.
		
	\item
		For each $j \in \MC{D}$ and each $i \in \MC{F}$, there is an arc $(j^s,i)$ of capacity $x_{i,j}$, an arc $(i,j^s)$ of capacity $g_{i,j}$, and an arc $(i^t,j^t)$ of capacity $r^{(g)}_j \cdot y_i$.
\end{itemize}
\end{definition}

%
For any $i \in \MC{F}, j \in \MC{D}$, let $P^{(g)}_{(x,y)}(i,j)$ to denote the set of paths in $\mathbf{MFN}_\Psi(x,y,g)$ for commodity $j$ to sink via $i^t$. 
%
The superscript $(g)$ and the subscript $(x,y)$ is omitted when there is no confusion in the context.
%
Let $\vphantom{\text{\Large T}} P := \cup_{i \in \MC{F}, j \in \MC{D}} P(i,j)$ to denote the set of possible paths.
%
See also Figure~\ref{cons-MFN-g} for an illustration on the construction of $\mathbf{MFN}_\Psi(x,y,g)$ and the corresponding constraints.
%
%The constraints of $\mathbf{MFN}_\Psi(x,y,g)$ is given in Figure~\ref{cons-MFN-g}.
%


%\smallskip

%
%An, Singh, and Svensson showed 
%in the following lemma 
%that, with all valid integral partial assignments taken into account, MFN gives a valid strong LP relaxation with constant integrality gap for CFL.
%

%\smallskip

%

%


%
\begin{figure*}[t]
%\centering
%
\begin{minipage}{\textwidth}
\fbox{
\enskip
\begin{minipage}{.39\textwidth}
\centering
\includegraphics[scale=1]{MFN-construction.pdf}
%\vspace{-12pt}
\end{minipage}
\enskip
}
%
\enskip
%
\fbox{
\hspace{-10pt}
\begin{minipage}{.51\textwidth}
%\vspace{-2pt}
\begin{subequations}
\begin{align}
& \sum_{i \in \MC{F}, \; p \in P(i,j)} \hspace{-4pt} f_p \; \ge \; r^{(g)}_j, & & \forall j \in \MC{D}, \label{LP_MFN_1} \\[6pt]
& \sum_{p \in P \colon (j^s,i) \in p} \hspace{-6pt} f_p \; \le \; x_{i,j}, & & \hspace{-8pt} \forall i \in \MC{F}, j \in \MC{D}, \label{LP_MFN_2} \\[6pt]
& \sum_{p \in P \colon (i,j^s) \in p} \hspace{-6pt} f_p \; \le \; g_{i,j}, & & \hspace{-8pt} \forall i \in \MC{F}, j \in \MC{D}, \label{LP_MFN_3} \\[6pt]
& \sum_{j \in \MC{D}, \; p \in P(i,j)} \hspace{-10pt} f_p \; \le \; u^{(g)}_i \hspace{-4pt} \cdot y_i, & & \forall i \in \MC{F}, \label{LP_MFN_4} \\[6pt]
& \sum_{p \in P(i,j)} f_p \; \le \; r^{(g)}_j \hspace{-4pt} \cdot y_i, & & \hspace{-8pt} \forall i \in \MC{F}, j \in \MC{D},  \label{LP_MFN_5} \\[6pt]
& \hspace{10pt} f_p \; \ge \; 0, & & \forall p \in P. \label{LP_MFN_6}
\end{align}
\end{subequations}
\vspace{-8pt}
\end{minipage}
\enskip
}
\end{minipage}
\caption{The construction of $\mathbf{MFN}_\Psi(x,y,g)$ and the corresponding LP constraints.}
\label{cons-MFN-g}
\vspace{-4pt}
\end{figure*}
%
%
%
%
\begin{lemma}[An, Singh, Svensson~\cite{DBLP:conf/focs/AnSS14}] \label{lemma-an-MFN-separation}
%
Given an instance $\Psi = (\MC{F},\MC{D}, \MB{c}, \MB{o}, \MB{u})$ of CFL,
% and $(x,y)$, the separation problem for the feasibility of $\mathbf{MFN}_\Psi(x,y,g)$ can be answered in weakly polynomial-time for any valid $g$.
the constraints defined by
$\mathbf{MFN}_\Psi(x,y) := \left\{ \; \mathbf{MFN}_\Psi(x,y,g) \text{ feasible} \; \colon \; \text{$\forall$ valid $g$} \; \right\}$
%\begin{align*}
%\mathbf{MFN}_\Psi(x,y) \; & := \; \left\{ \; \vphantom{\text{\LARGE T}} \mathbf{MFN}_\Psi(x,y,g) \text{ feasible} \; \colon \; \text{$\forall$ valid $g$} \; \right\} 
%\\
%& \; = \; \left\{ \; \vphantom{\text{\LARGE T}} \mathbf{MFN}_\Psi(x,y,g) \text{ feasible} \; \colon \; \text{$\forall$ valid integral $g$} \; \right\}
%\end{align*}
is a valid relaxation for integral solutions on $\Psi$.
%
Furthermore, 
%the polytope 
the separation problem for the feasibility of $\mathbf{MFN}_\Psi(x,y,g)$ for any valid $g$ can be answered in weakly polynomial-time, and a basic feasible flow can be obtained.
%
\end{lemma}

%\begin{definition}[Supporting Flow]
%
%A feasible flow $f \colon P \rightarrow [0,1]$ for an $\mathbf{MFN}_\Psi(x,y,g)$ network is said to be $\alpha$-supporting for some $0 < \alpha < 1$ if
%$$\sum_{i \in S_y(\alpha)} \; \sum_{p \in P_{i,j}} f_p \ge 1-\alpha \quad \text{holds for all $j \in \MC{D}$, }$$ 
%where $S_y(\alpha)$ is defined as $S_y(\alpha) := \left\{ \; i \in \MC{F} \; \colon \; y_i < 1-\alpha \; \right\}$.
%\end{definition}

%

%The following lemma proves the existence of a nice flow, provided that the partial assignment evolving the network is carefully chosen.
%
%Intuitively, it states that, in the specific flow each commodity will send at least half of its demand to the small facilities.
%\mong{***** blahblah *****}

%

%\begin{lemma}[An et al.~\cite{DBLP:conf/focs/AnSS14}]
%\label{an-lemma-cfl-reduction-to-half-integral-solutions}
%
%Given an instance $\Psi = (\MC{F}, \MC{D}, \MB{o}, \MB{c}, \MB{u})$ of CFL, a candidate $(x,y)$ where $x \in [0,1]^{\MC{F} \times \MC{D}}, y \in [0,1]^{\MC{F}}$, in polynomial time we can compute a $(1/2,y')$-concentrated partial assignment $\MB{g}$ with $\MB{g} \le 2x$, where $y'$ is defined as
%$$y'_i := \begin{cases}
%\; 1, & \text{if $y_i \ge 1/4$,} \\
%\; y_i, & \text{otherwise,}
%\end{cases}$$
%such that the feasibility of $\mathbf{MFN}_\Psi(x,y',g)$ implies the existence of a $1/2$-supporting flow within.
%is infeasible, 
%		%thereby showing the infeasibility of $(x,y)$ for $\mathbf{MFN}(\Psi)$, 
%		or
%		
%	\item
%		(b) compute a $y'$-concentrated partial assignment $\MB{g}$ with $\MB{g} \le 2x$, where $y'$ is defined as
%		$$y'_i := \begin{cases}
%		\; 1, & \text{if $y_i \ge 1/4$,} \\
%		\; y_i, & \text{otherwise.}
%		\end{cases}$$
%		and a feasible flow $\MB{f}$ for $\mathbf{MFN}_\Psi(x,y',g)$ that is $1/2$-supporting.
%		%
%		\mong{*****}
%\end{enumerate}
%\end{lemma}

%

%

%

%

%

%

%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%

\section{LP-based Approximation for CFL}
\label{sec-approx-cfl}

%\mong{*** $0 < \alpha < 1$ parameter to be decided ***}

%
In this section we describe our approximation algorithm for CFL and prove Theorem~\ref{thm-approx-cfl}.
%
%\mong{**** important **** $0 < \alpha \le 1/2$ *****}
%
%\mong{****** $0 < \alpha < 1/3$ ******* important ******}
%
%
%\smallskip
%
%
Our algorithm inherits the round-or-separate framework used in~\cite{DBLP:conf/focs/AnSS14} for producing candidate solutions that can either be rounded or separated.
%
For completeness we first describe the details of the framework in the following.
%
Then we present our modified construction for obtaining sparsely-loaded flow and our iterative rounding scheme.
%

%\smallskip

\vspace{-10pt}

%
\paragraph{The Round-or-Separate Framework.}

Let $\Psi = (\MC{F}, \MC{D}, \MB{o}, \MB{c}, \MB{u})$ be an instance of CFL. 
%
The algorithm starts by guessing the cost of the optimal 
%
%
solution using a standard binary search.
%
For each guess, say, $\gamma$, the algorithm applies the 
%
Ellipsoid algorithm on~LP-(MFN)
%\ref{LP-MFN} 
%given 
%below 
in Figure~\ref{fig-LP-MFN} with cost~$\gamma$.
%
%
%\smallskip
%
%
For each separation problem incurred in the Ellipsoid algorithm,
% on the candidate solution, 
say, for $(\bm{x}', \bm{y}')$, we apply Theorem~\ref{thm-MFN-relaxed-separation-oracle},
%, 
which is stated below,
%which is given below and 
%and to be proved in the rest of this section, 
for either a 
%violated constraint
separating hyperplane or an integral solution $\vphantom{\text{\LARGE T}} \left(\bm{x}^{(\gamma)}, \bm{y}^{(\gamma)}\right)$
%, rounded from $(\bm{x}', \bm{y}')$, 
with the claimed approximation guarantee.
%

\smallskip

%
%
\begin{wrapfigure}[8]{r}{.34\textwidth}
%\begin{figure*}[t]
\begin{flushright}
\begin{minipage}{.33\textwidth}
%
\vspace{-26pt}
\fbox{
\hspace{1pt}
\begin{minipage}{.86\textwidth}
%\begin{subequations}
\vspace{-6pt}
\begin{align*}
& \mathbf{MFN}_\Psi(x,y), \\[4pt]
& \psi(\bm{x}, \bm{y}) \le \gamma,  \notag \\[5pt]
& \bm{x} \in [ \hspace{1pt} 0, 1 \hspace{1pt} ]^{\MC{F} \times \MC{D}}, \; \bm{y} \in [ \hspace{1pt} 0,1 \hspace{1pt} ]^{\MC{F}}. \notag
\end{align*}
\vspace{-12pt}
\end{minipage}\hspace{1pt}
}
%\tag*{LP-(MFN)} \label{LP-MFN}
\caption{LP-(MFN). }
%\caption{The decision version of MFN relaxation for CFL.}
\label{fig-LP-MFN}
\end{minipage}
\end{flushright}
%\end{figure*}
\end{wrapfigure}
%
When an integral solution is successfully rounded or when the Ellipsoid algorithm concludes the infeasibility of $(\bm{x}', \bm{y}')$, the algorithm proceeds to the next iteration of the binary search process until the desired precision is attained.
%
When the binary search process terminates, the algorithm outputs $\left(\bm{x}^{(\gamma_0)}, \bm{y}^{(\gamma_0)}\right)$ for the smallest $\gamma_0$ for which an integral solution is successfully rounded when $\gamma_0$ is considered.
%

\smallskip

%
Our result for CFL is the following theorem.
%\pagebreak

\begin{theorem}	\label{thm-MFN-relaxed-separation-oracle}
%
Given 
%target parameter $\gamma$ and 
a candidate solution $(\bm{x}', \bm{y}')$ for~LP-(MFN)
%\ref{LP-MFN} 
and a target parameter $\alpha$ with $0 < \alpha \le 1/3$, we can compute in polynomial-time either (i) a 
%violated constraint for
separating hyperplane for $(\bm{x}', \bm{y}')$
% and the feasible region of
and~LP-(MFN),
%\ref{LP-MFN}, 
or (ii) an integral solution $(\bm{x}^*, \bm{y}^*)$ rounded from $(\bm{x}', \bm{y}')$ for $\Psi$ with 
%$\psi(\bm{x}^*, \bm{y}^*) \le ??\cdot \gamma$. \mong{*** pin down the ratio ***}
%
$$\psi(x^*, y^*) \le \max\left\{ \; \frac{3}{2\alpha} \; , \; \frac{7-4\alpha}{(1-\alpha)^2} \; \right\} \times \psi(x',y').$$
\end{theorem}

%
To prove Theorem~\ref{thm-approx-cfl}, it suffices to prove the statement of Theorem~\ref{thm-MFN-relaxed-separation-oracle} and observe that the specific choice of $\alpha := \left(10-\sqrt{67}\right)/11$ gives the claimed result.
%
We also note that, it suffices to consider the MFN-type constraints as 
%the validation of 
the remaining constraints in~LP-(MFN) can be verified directly.
%

\smallskip

%
In the rest of this section, we describe our iterative rounding process 
%for $(\bm{x}', \bm{y}')$ that completes the proof of
that establishes
%for 
Theorem~\ref{thm-MFN-relaxed-separation-oracle}.
%
We complete the proof in Section~\ref{sec-proof-approx-cfl}, page~\pageref{sec-proof-approx-cfl}.


%

%%%
%%%

%\paragraph{Rounding Process.}
\paragraph{Obtaining a Sparsely-Loaded Flow or a Separating Hyperplane.}

Let $(\bm{x}',\bm{y}')$ be a candidate solution and $0 < \alpha \le 1/3$ be the target parameter.
%
We classify the facilities as follows.
%start by classifying the facilities.
%
Let $I := \left\{ \; i \in \MC{F} \; \colon \; 0 < y'_i < \alpha \; \right\}$
% \quad \text{and} \quad 
and $U := \left\{ \; i \in \MC{F} \; \colon \; y'_i \ge \alpha \; \right\}$.
%
The facilities in $U$ are further classified into two categories. Let 
\vspace{-8pt}
$$\hspace{0.2cm} U^{(>)} := \left\{ \; i \in U \; \colon \; \sum_{j \in \MC{D}} x'_{i,j} \; > \; (1-\alpha)\cdot u_i  \;\; \right\}$$
and $U^{(\le)} := U \setminus U^{(>)}$.
%
Provided that the facilities in $U$ are to be rounded up in the approximate solution, we know that the assignments to $U^{(\le)}$ are ready for rounded up as well.
%

\smallskip

%
Consider the maximum b-matching problem in the bipartite graph $G = \left(\MC{D},U^{(>)},E\right)$, 
%formed from $\MC{D}$ to $U$ whose edges 
where for each $j \in \MC{D}$ and $i \in U^{(>)}$, there exists an edge $(j,i)$ in $E$ with
%are further associated with capacity $(1+\alpha)\cdot\left.\bm{x}'\right|_{\MC{D},U}$.
%
%In particular, each edge $(j,i)$ between $j \in \MC{D}$ and $i \in U$ has 
capacity $x'_{i,j} / (1-\alpha)$.
%
Solve the maximum b-matching problem on $G$ for an optimal assignment $\bm{h}$.
%

%\medskip
\smallskip

%
We say that a client $j \in \MC{D}$ is partially-assigned in $\bm{h}$ 
if $\sum_{i \in U^{(>)}} h_{i,j} < 1$ and fully-assigned otherwise.
%
%Consider the flow-network to which $\bm{h}$ corresponds and any $j \in \MC{D}$ that is partially-assigned.
% in $\bm{h}$.
%
%\begin{definition}[Augmenting Path]
%Let $j \in \MC{D}$ be partially-served in $\bm{h}$.
%
We say that an alternating path $P$ between $\MC{D}$ and $U^{(>)}$ that starts at a partially-assigned client $j \in \MC{D}$ is an \emph{augmenting path} if the following holds.
\begin{itemize}
	\item
		For each $(j',i') \in P$, where $j' \in \MC{D}, i'\in U^{(>)}$, we have $h_{i',j'} < x'_{i,j} / (1-\alpha)$.
	\item
		For each $(i',j') \in P$, where $i' \in U^{(>)}, j' \in \MC{D}$, we have $h_{i',j'} > 0$.
\end{itemize}
%\end{definition}
%
\noindent
Intuitively, an augmenting path that starts from a partially-assigned $j$ is a way of increasing its assignment without altering the optimality of maximum b-matching between $\MC{D}$ and $U^{(>)}$.
% i.e., each edge of $P$ has nonzero tolerance in modifying the flow $\bm{h}$.

\smallskip

%
%We need the following concept.
%\begin{definition}
We 
%further 
say that a facility $i \in U^{(>)}$ is {tightly-occupied} if there exists an augmenting path 
%that connects
between a partially-assigned client 
%to it.
and $i$.
% that starts at a partially-assigned client.
%$\sum_{j \in \MC{D}} h_{i,j} = u_i$ and {unoccupied} otherwise.
%
%We say that a client $j \in \MC{D}$ is loosely-assigned if $\sum_{i \in U^{(>)}} h_{i,j} = r_j$ and 
% $\vphantom{\text{\Large T}} j \in \MC{D}$ is \emph{tightly-assigned} if 
%$h_{i,j} < x'_{i,j} / (1-\alpha)$ holds for some $i \in U^{(>)}$ that is unoccupied.
%\end{definition}
%
%
%\medskip
%
%
%\noindent
%
%\smallskip
%
%
For each $i \in \MC{F}, \; j \in \MC{D}$, define the partial assignment $g_{i,j}$ as
% follows.
%
$$g_{i,j} := \begin{cases}
\; h_{i,j}, & \text{if $i \in U^{(>)}$ is tightly-occupied,} \\
\; 0, & \text{otherwise.}
\end{cases}$$
%\; 0, & \text{if $i \in U^{(\le)}$, or, $i \in U^{(>)}$ is partially-occupied and $j$ is tightly-assigned,} \\
%\; h_{i,j}, & \text{otherwise.}
%
%\; h_{i,j}, & \text{if $i \in U^{(\ge)}$ is fully-occupied,} \\
%\; h_{i,j}, & \text{if $i \in U^{(\ge)}$ is partially-occupied and $j$ is not tightly-assigned,} \\
%\; 0, & \text{otherwise.}
%
Apply Lemma~\ref{lemma-an-MFN-separation} for either a basic feasible flow $\bm{f}$ or a 
%violated constraint
separating hyperplane for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, where $y''_i := 1$ for all $i \in U$ and $y''_i := y'_i$ otherwise.
%
If a violated constraint is found, report it to the Ellipsoid algorithm.
% and proceed to the next iteration.
%
Otherwise, we proceed to round the flow $\bm{f}$.
%

\smallskip

%
%Note that, 
%s
Since feasible solutions of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}',\bm{g})$ are also feasible for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, hyperplanes separating $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ 
also separate $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}',\bm{g})$.
% \mong{***} 
Furthermore, since $\bm{f}$ is a basic feasible solution,
% for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, 
%which has a polynomial number of constraints in $\MC{F}$ and $\MC{D}$, 
%whose number of constraints is polynomial in $\MC{F}$ and $\MC{D}$, 
it follows that 
%the number of nonzero coordinates in $\bm{f}$
%, i.e., 
the number of paths on which nonzero flow are sent in $\bm{f}$
is at most the number of constraints in $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, which is 
%also 
polynomial in $\MC{F}$ and $\MC{D}$.
%\mong{*** solve for a basic solution *** so that $\bm{f}$ is of polynomial complexity ***}

%

%


%%%
%%%

\paragraph{Rounding the Small Instance via Sparsely-Loaded Flow.}

In the following we describe our iterative rounding process for facilities in $I$ 
%and the assignments made to them 
using the flow $\bm{f}$ obtained in the previous step.
%

\smallskip

%
During the rounding process, the algorithm maintains a parameter tuple $\Psi' = (F', D', \bm{r}')$ which denotes the remaining instance to be processed,
%.
%
%Intuitively, $F'$ and $D'$ denote the set of facilities and the set of clients yet to be processed, and 
where $\bm{r}'$ 
%denotes
is the residue demand of each $j \in D'$.
% that needs to be assigned.
%
The algorithm updates the parameter tuple gradually during the 
%iterations of the 
process until $D'$ becomes empty.
%

\smallskip

%
Initially, for each $j \in \MC{D}$, the residue demand of $j$ is defined to be $r'_j := \sum_{i \in I, \; p \in P'(i,j)} f_p,$
where $P'(i,j)$ is the set of paths for which nonzero flow for commodity $j$ is sent to sink via $i$ in $\bm{f}$.
%, i.e., the paths that starts from $j^s$ to $j^t$ via $i^t$.
% and that carries nonzero flow for commodity $j$ in $\bm{f}$.
%
%Let $F'$ and $D'$ be the set of facilities and the set of clients yet to be processed.
%
%Initially 
The algorithm sets $F' := I$ and 
$D' := \{ \; j \in \MC{D} \; \colon \; r'_j \; > \; \alpha \cdot r_j \; \}$, 
%\quad \text{where} \enskip
where $r_j := 1 - \sum_{i \in U^{(>)}} g_{i,j}$ 
is the demand to be sent in $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ for 
%any 
commodity $j \in \MC{D}$.
%
%Intuitively, the set $D'$ consists of clients which have sent at least $\alpha$ fraction of its demand to facilities in $F'$.
%

\smallskip

%
%The following notion regarding \emph{per-unit-assignment rerouting cost} is key in 
%establishing the approximation guarantee of 
%our rounding process.
%s in our algorithm.
%
%Let $(\bm{x}, \bm{y})$ be an arbitrary solution of interests.
% and $\bm{\lambda}$ be
%assignment function, a multiplicity function, and 
%an estimated assignment radius of the clients.
% of interests.
For any $i \in F'$ with $\sum_{j \in D'} x_{i,j} > 0$ and any 
%solution 
$(\bm{x},\bm{y})$ of interests, define the per-unit-assignment rerouting cost 
%(from the vicinity of $i$) 
to $i$ as 
%\mong{**** $F'$ not required ****}
\vspace{-4pt}
\begin{align*}
\theta(i,D',\bm{x}, \bm{y}) \; := \; \frac{1}{\sum_{j \in D'} x_{i,j}} \cdot \left( \;\; 3\cdot o_i\cdot y_i \; + \; 2\cdot \sum_{j \in D'} c_{i,j} \cdot x_{i,j}  \;\; \right).
\end{align*}
%where $\bm{x}$, $\bm{y}$, and $\bm{\lambda}$ are the assignment function, the multiplicity function, and the estimated assignment radius of interests.
%
Provided that $(\bm{x},\bm{y})$ is optimal for~\ref{LP-natural-CFL-modified} given in Figure~\ref{fig-cfl-natural-modified-main-content}, 
%Intuitively, 
%by 
standard complementary slackness 
%conditions, it follows
guarantees that $c_{i,j} \le \lambda_j$ for all $x_{i,j}>0$, where
$\bm{\lambda}$ comes from an optimal solution for~\ref{LP-natural-modified-dual}.
%
Furthermore, it follows that $3\cdot o_i\cdot y_i + 2\cdot \sum_{j \in D'} c_{i,j} \cdot x_{i,j} = o_i\cdot y_i + 2\cdot \sum_{j \in D'} \lambda_j \cdot x_{i,j}$,
%, where
%$\bm{\lambda}$ comes from an optimal solution for~\ref{LP-natural-modified-dual},
%
and $\vphantom{\text{\Large T}} \theta(i,D',x,y)$ 
%thereby 
corresponds implicitly and naturally to the cost incurred if we reroute the assignments which are originally assigned to 
%the
%from 
facilities in the vicinity of $i$ simultaneously and proportionally to $i$.
% instead.
% in proportion to the current flow assignments.
%

\smallskip

%
%\begin{wrapfigure}[28]{r}{.5\textwidth}
\begin{figure*}[h]
%\centering
%
\begin{minipage}{.42\textwidth}
\fbox{
\begin{minipage}{\textwidth}
\vspace{-8pt}
\begin{align}
& \text{Minimize} \enskip \psi(x,y)
%\sum_{i \in F'} o_i \cdot y_i \; + \; \sum_{i \in F', \; j \in D'} c_{i,j}\cdot x_{i,j} 
& &  \label{LP-natural-CFL-modified} \tag*{LP-(M)} \\[6pt]
& \sum_{i\in F'} \; x_{i,j} \; = \; r'_j, & & \forall j\in D',  \notag \\[4pt]
& \sum_{j \in D'} \; x_{i,j} \; \le \; u_i \cdot y_i, & & \forall i \in F',  \notag \\[1pt]
& x_{i,j} \le \frac{2\alpha}{1-\alpha} \cdot r_j \cdot y_i, & & \forall i \in F', j \in D',  \notag \\[4pt]
& y_i \; \le \; \frac{1-\alpha}{2}, & & \forall i \in F',  \notag \\[4pt]
& x_{i,j} \ge 0, \; y_i \ge 0, & & \forall i \in F', j \in D'. \notag
\end{align}
\vspace{-12pt}
\end{minipage}
}
\end{minipage}
%
\quad\enskip
%
\begin{minipage}{.5\textwidth}
\fbox{
\begin{minipage}{\textwidth}
\vspace{-10pt}
\begin{align}
& \label{LP-natural-modified-dual} \tag*{LP-(DM)} \\[8pt]
& \text{Maximize} \enskip \sum_{j \in D'} r'_j \cdot \lambda_j \; - \; \sum_{i \in F'} \; \frac{1}{2} \cdot (1-\alpha) \cdot \eta_i \notag \\[6pt]
& \lambda_j \; \le \; \beta_i \; + \; \Gamma_{i,j} \; + \; c_{i,j}, \; \qquad\enskip \forall i \in F', j\in D',  \notag \\[6pt]
& u_i \cdot \beta_i \; + \; \frac{2\alpha}{1-\alpha} \cdot \sum_{j \in D'} r_j \cdot \Gamma_{i,j} \; \le \; o_i \; + \; \eta_i, \notag \\[-4pt]
& \hspace{4.9cm} \quad  \forall i \in F',  \notag \\[8pt]
& \lambda_j \in \MBB{R}, \; \beta_i, \; \Gamma_{i,j}, \; \eta_i \; \ge \; 0,  \; \qquad\enskip \forall i \in F', j \in D'.  \notag
%\tag*{(DM-3)}
\end{align}
\vspace{-6pt}
\end{minipage}
}
\end{minipage}
%
\caption{The natural LP formulations for our iterative rounding process.}
\label{fig-cfl-natural-modified-main-content}
%
%\caption{The dual LP of \ref{LP-natural-CFL}. \mong{*******}}
%\label{fig-cfl-natural-modified-dual}
\end{figure*}
%\end{wrapfigure}
%


%
In the following, we describe the rounding process in details.
%
In each iteration, the algorithm solves~\ref{LP-natural-CFL-modified}
% and its dual LP 
%given in Figure~\ref{fig-cfl-natural-modified-main-content} 
on the current parameter tuple $\Psi'$ for optimal 
%primal and dual solutions 
$(\bm{x}^\dagger, \bm{y}^\dagger)$
% and $(\bm{\lambda}^\dagger, \bm{\beta}^\dagger, \bm{\Gamma}^\dagger, \bm{\eta}^\dagger)$. 
%
%Note that, by complementary slackness it follows that $\lambda^\dagger_j \ge c_{i,j}$ for all $i \in F', j \in D'$ with $x^\dagger_{i,j} > 0$.
%
%
%\smallskip
%
%
%The algorithm then
and selects a facility $i \in F'$ to form a cluster.
% as follows.
%
Depending on the status of the facilities in $F'$, we have two cases.
%
%\smallskip
%
\begin{itemize}
	\item
		If $y^{\dagger}_i = (1-\alpha)/2$ for some $i \in F'$, 
		%
		%In this case, 
		then the algorithm picks an arbitrary $i$ with $y^{\dagger}_i = (1-\alpha)/2$ and sets $\sigma^{(i)}_k = 0$ for all $k \in F' \setminus \{i\}$.
		

	\item
		If $y{\dagger}_i < (1-\alpha)/2$ for all $i \in F'$,
		%, then
		%
		%
		%In this case,
		then the algorithm selects among the facilities $i\in F'$ with $\sum_{j \in D'} x^\dagger_{i,j} > 0$, 
		%\mong{*** $y^{\dagger}_i > 0$ *** important ******}
		%the algorithm selects a 
		the facility $i$ with the minimum $\theta(i,D',\bm{x}^\dagger,\bm{y}^\dagger)$.
		% to form a cluster.
%, and rounds its facility value to $\alpha$ by multiplying it by a factor of $\alpha \frac{1}{y_i}$.
%
Intuitively, the facility $i$ has the lowest per-unit-assignment rerouting cost, and any other facility in $F'$ can afford the rerouting cost within the cluster centered at $i$.
%

\smallskip

%
%\mong{*** $r_j$ ***}
%
For each $j \in D'$, let 
%$$\delta^{(i)}_j \; := \; \alpha \cdot \frac{1}{y'_i} \cdot \sum_{p \in P(i,j)} f_p$$
$\delta^{(i)}_j := \left( \; (1-\alpha) / (2 y^\dagger_i) - 1 \; \right) \cdot x^\dagger_{i,j}$ 
denote the amount of assignment to be gathered from the vicinity of facility $i$ to $i$ via client $j$.
%

\smallskip

%
For each $k \in F' \setminus \{i\}$, define the contribution (fraction) of $k$ towards $i$ to be
$$\sigma^{(i)}_k \; := \; \frac{1}{\sum_{\ell \in D'} x^\dagger_{k,\ell}} \cdot \sum_{j \in D'} \sigma^{(i)}_{k,j}, \quad \text{where} \enskip \sigma^{(i)}_{k,j} \; := \; \frac{x^\dagger_{k,j}}{\sum_{\ell \in F' \setminus \{i\}} x^\dagger_{\ell,j}} \cdot \delta^{(i)}_j \enskip \text{is}$$
the contribution of $k$ to be sent via client $j$.
%
Note that, from the definition it follows that $\sum_{k \in F' \setminus \{i\}} \sigma^{(i)}_{k,j} = \delta^{(i)}_j$, and the amount of flow to be gathered via $j$ can be fulfilled.
%
\end{itemize}

%\smallskip

%
% to be 
%$$\sigma_{k,j} \; := \; \sum_{p \in P(k,j)} f_p \frac{1}{\Delta_j}$$
%
%For each $k \in F' \setminus \{i\}$, the algorithm scales down $y'_k$ by $(1 - \sigma^{(i)}_k)$.
%
%For each $j \in D'$, the algorithm further scales down $x^\dagger_{i,j}$ by $(1-\sigma^{(i)}_k)$.
%

\medskip

%
\noindent
The algorithm updates the 
%remaining instance $(F', D', \bm{r}')$ 
parameter tuple $\Psi'$ as follows.
%
The algorithm 
%then 
%sets $y^*_i$ to be $1$ and 
removes $i$ from $F'$
% and updates $D'$ as follows.
and sets $r'_j := \sum_{k \in F'} \; (1-\sigma^{(i)}_k) \cdot x^\dagger_{k,j}$ for all $j \in D'$.
%
For each $j \in D'$ with $r'_j \le \alpha \cdot r_j$, the algorithm removes $j$ from $D'$.
% and sets $f_p$ to be zero for all $k \in F'$ and $p \in P(k,j)$.
%
Then the algorithm proceeds to the next iteration until $D'$ becomes empty.

%

%%%
%%%

\paragraph{Final Output.}

%
When $D'$ becomes empty, the algorithm forms the integral multiplicity function $\bm{y}^*$ by setting $y^*_i := 1$ for all $i \in U \cup \left( I \setminus F' \right)$ and $y^*_i := 0$ otherwise.
%
%\mong{****}
%Provided that the integral multiplicity function $y^*$ is formed, t
The algorithm solves the min-cost assignment problem on $\MC{D}$ and $\MC{F}^* := \left\{ \hspace{1pt} \vphantom{\text{\Large T}} i \in \MC{F} \; \colon \; y^*_i = 1 \hspace{1pt} \right\}$ for an optimal integral assignment $\bm{x}^*$ and 
%
%The algorithm 
reports $(\bm{x}^*, \bm{y}^*)$ as the claimed integral solution.


\smallskip

%
%\noindent
%\mong{****} A pseudo-code of this algorithm is provided in Figure~\ref{fig-overview-tight-vc-hc} in Page~\pageref{fig-overview-tight-vc-hc} for further reference.

%\smallskip

%\mong{****} 
%It is clear that the algorithm runs in polynomial time.
%
%The following lemma, which is proved in Section~\ref{sec-proof-approx-cfl}, page~\pageref{sec-proof-approx-cfl}, establishes the approximation guarantee 
%for $(\bm{x}^*, \bm{y}^*)$ 
%and completes the proof for Theorem~\ref{thm-MFN-relaxed-separation-oracle}.

%\begin{lemma} \label{theorem-cfl-sparse-flow-approx}
%Provided that $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ is feasible, the feasible region of the min-cost assignment problem on $\MC{D}$ and $F^*$ is non-empty.
%
%Furthermore, 
%\mong{**** ratio ****}
%\mong{***} Let $\Pi$ be the input instance of CFL-UFC and $(\bm{x}', \bm{y}')$ be an optimal solution for~\ref{LP-natural-CFL} on $\Pi$.
%
%Algorithm $\MC{A}$ computes a feasible integral solution $(x^\dagger, y^*)$ with
%$$\Psi(x^\dagger, y^*) \le 4\cdot \Psi(x',y').$$
%$$\psi(x^*, y^*) \le \max\left\{ \; \frac{3}{2\alpha} \; , \; \frac{7-4\alpha}{(1-\alpha)^2} \; \right\} \times \psi(x',y').$$
%\end{lemma}

%

%

%


%

%A natural LP relaxation for the CFL problem is formulated as follows.

%


%%%%%
%%%%%

\section{$4$-Approximation for CFL-CFC}
\label{sec-approx-cfl-cfc}

%

%In this section, we present our rounding-based approximation algorithm for CFL-UFC and prove Theorem~\ref{thm-approx-cfl-ufc}.
%
In the following, we describe our approximation algorithm $\MC{A}$ for CFL-CFC and prove Theorem~\ref{thm-approx-cfl-ufc}.
%
Let $\Psi = (\MC{F},\MC{D}, \MB{c}, \MB{u})$ 
%with $\MB{o} = 1$ 
be an instance of CFL-CFC.
%

\smallskip

%
For the ease of presentation, for any $A \subseteq \MC{F}$, any assignment $x$, and any $j \in \MC{D}$, we 
%will 
use $N_{(A,x)}(j) := \left\{ \; \vphantom{\text{\Large T}} i \in A \; \colon \; x_{i,j} > 0 \; \right\}$ to denote the set of facilities in $A$ to which $j$ is assigned to in $x$.
%
Similarly, for any $B \subseteq \MC{D}$ and any $i \in \MC{F}$, we 
%will 
use $N_{(B,x)}(i) := \left\{ \; \vphantom{\text{\Large T}} j \in B \; \colon \; x_{i,j} > 0 \; \right\}$ to denote the set of clients in $B$ that is assigned to $i$ in $x$.
%

\smallskip

%
Solve~\ref{LP-natural-CFL} and its dual LP, given below in Figure~\ref{fig-cfl-cfc-natural-lp}, on $\Psi$ for optimal primal and dual solutions $(\bm{x}', \bm{y}')$ and
%$(\bm{x}', \bm{y}')$ be an optimal solution for~\ref{LP-natural-CFL} on $\Pi$ and 
$(\bm{\alpha}, \bm{\beta}, \bm{\Gamma}, \bm{\eta})$.
% be an optimal solution for the dual LP of~\ref{LP-natural-CFL}, given above in Figure~\ref{fig-cfl-cfc-natural-lp}.
%, for a witnessing dual solution $(\alpha, \beta, \Gamma, \eta)$.
%
It follows that, $\alpha_j \ge c_{i,j}$ for each $i\in \MC{F}, j \in \MC{D}$ with $x'_{i,j} > 0$.
%
We will use the fact that $\alpha_j$ is a valid estimation on the assignment radius for each $j \in \MC{D}$ in $\bm{x}'$.
%

%

%
%\begin{wrapfigure}[28]{r}{.5\textwidth}
\begin{figure*}[h]
%\centering
%
\begin{minipage}{.44\textwidth}
\fbox{\hspace{-2pt}
\begin{minipage}{\textwidth}
\vspace{-8pt}
\begin{align}
& \text{min} \; \sum_{i \in \MC{F}} \; y_i + \hspace{-2pt} \sum_{i \in \MC{F}, j \in \MC{D}} c_{i,j} \cdot x_{i,j} & &  \label{LP-natural-CFL} \tag*{LP-(N)}  \\[2pt]
& \; \sum_{i\in \MC{F}} \; x_{i,j} \; \ge \; 1, & & \hspace{-1.2cm} \forall j\in \MC{D}, 
% \tag*{(N-1)} 
\notag \\
& \; \sum_{j \in \MC{D}} \; x_{i,j} \; \le \; u_i \cdot y_i, & & \hspace{-1.2cm}  \forall i \in \MC{F},  
%\label{cons-P-2} \tag*{(N-2)} 
\notag \\
& \;\; 0 \; \le \; x_{i,j} \le y_i,   & & \hspace{-1.2cm}   \forall i \in \MC{F}, j \in \MC{D}, 
% \label{cons-P-3} \tag*{(N-3)} 
\notag \\[2pt]
& \;\; 0 \; \le \; y_i \; \le \; 1,   & & \hspace{-1.2cm}   \forall i \in \MC{F}, \notag
% \label{cons-P-4} \tag*{(N-4)}
\end{align}
\vspace{-14pt}
\end{minipage}
}
\end{minipage}
%
\quad\hspace{-2pt}
%
\begin{minipage}{.48\textwidth}
\fbox{\hspace{1pt}
\begin{minipage}{\textwidth}
%\vspace{-10pt}
\begin{align}
& \text{max} \enskip \sum_{j \in \MC{D}} \alpha_j \; - \; \sum_{i \in \MC{F}} \; \eta_i  & & \label{LP-natural-dual} \tag*{LP-(DN)} \\[4pt]
& \alpha_j \; \le \; \beta_i \; + \; \Gamma_{i,j} \; + \; c_{i,j},  & &  \forall i \in \MC{F}, j\in \MC{D},  \notag \\[6pt]
& u_i \cdot \beta_i \; + \; \sum_{j \in \MC{D}} \Gamma_{i,j} \; \le \; 1 \; + \; \eta_i,  & & \forall i \in \MC{F},  \notag \\[6pt]
& \alpha_j, \; \beta_i, \; \Gamma_{i,j}, \; \eta_i \; \ge \; 0,  & & \forall i \in \MC{F}, j \in \MC{D}.  \notag
%\tag*{(DM-3)}
\end{align}
\vspace{-6pt}
\end{minipage}
}
\end{minipage}
%
\caption{The natural LP formulations for CFL-CFC.}
\label{fig-cfl-cfc-natural-lp}
%
%\caption{The dual LP of \ref{LP-natural-CFL}. \mong{*******}}
%\label{fig-cfl-natural-modified-dual}
\end{figure*}
%\end{wrapfigure}
%


%\smallskip

%

%%%
%%%

%\subsection{The Rounding Algorithm}

%
%In the following we describe our rounding process in details.
%
%Let $\Pi = (\MC{F},\MC{D}, \MB{c}, \MB{u})$ be the input instance of CFL-UFC.
%
%The algorithm first solves~\ref{LP-natural-CFL-UFC} on $\Pi$ for an optimal solution $(\bm{x}', \bm{y}')$.

%


%\smallskip



%

%%%%
%%%%

%\subsection{Preliminary Rounding and Reduction to Half-Integral CFL}
%\paragraph{Preliminary Rounding and Classification.}
%\subsection{Initial Classification and Reduction.}
\paragraph{Initial Classification.}

%
Let $I := \left\{ \; i \in \MC{F} \; \colon \; 0 < y'_i < \frac{1}{2} \; \right\}$
and
% \quad \text{and} \quad 
$U := \left\{ \; i \in \MC{F} \; \colon \; y'_i \ge \frac{1}{2} \; \right\}.$
%
%
%\smallskip
%
%
%The algorithm rounds up $y'_i$ for all $i \in U$.
%
The clients in $\MC{D}$ are divided into three categories, namely, those that are served merely by $I$, those that are served jointly by $I$ and $U$, and those that are served merely by $U$.
%
Let $$J^{(I)} := \left\{ \vphantom{\text{\LARGE T}} \; j \in \MC{D} \hspace{1pt} \colon \hspace{1pt} x'_{i,j} = 0 \enskip \text{for all $i \in U$} \; \right\}, \enskip J^{(\leftrightarrow)} := \left\{ \; j \in \MC{D} \hspace{1pt} \colon \min\left( \max_{i \in I}x'_{i,j} \hspace{2pt} , \hspace{2pt} \max_{i \in U}x'_{i,j} \right) > 0 \; \right\},$$
and $J^{(U)} := \MC{D} \setminus \left( J^{(I)} \cup J^{(\leftrightarrow)} \right)$ denote the clients in the three categories, respectively.
%
Note that, in addition to the rounding problem for facilities in $I$, we need to resolve the rounding problem for assignments of the clients in the first two categories as well.
%


%\medskip
%\bigskip

%\smallskip

\paragraph{Our Rounding Process.}

%
%Let $y^*$ denote the integral multiplicity function our rounding algorithm will form.
%
%Initially $y^*_i := 1$ for $i \in U$ and $y^*_i := 0$ otherwise.
% and
Let $F'$ and $D'$ be the set of facilities and the set of clients yet to be processed, and $x^*$ be an intermediate assignment function our algorithm will maintain during the process.
%
Initially, $F' := I$, $D' := J^{(I)} \cup J^{(\leftrightarrow)}$, and $x^* := 0$.
%, and  Initially $F' := \MC{F}$, $D' := \MC{D}$, and $y^* := 0$.
%

%\smallskip

%
%The following concept is key to our rounding process. 
%
%\mong{**** remove this concept ****}

%\begin{definition}
%The assignment 
%function 
%$x'$ is said to be supporting if $\sum_{i \in F'}x'_{i,j} \ge 1/2$ for all $j \in D'$.
%\end{definition}

%
%Intuitively, when the assignment $x'$ is supporting, it follows from~\ref{cons-P-3} in~\ref{LP-natural-CFL} that, for each $j \in D'$, the total facility value in $N_{(I \cap F',x')}(j)$ is well-bounded from below.
%

\smallskip

%
Our rounding algorithm consists of two phases.
%
In the first phase, it proceeds in iterations to form clusters.
%
In each of such iterations, the algorithm checks if 
$\sum_{i \in F'} x'_{i,j} \ge 1/2$ holds for all $j \in D'$.
%the current assignment $x'$ is supporting.
%
If not, the algorithm makes it so by repeatedly removing small clients from $D' \cap J^{(\leftrightarrow)}$ and redistributing their demand to facilities in $U$ to form a set of \emph{outlier clients}.
% that are located at facilities in $U$.
%
We use $H$ to denote the set of outlier clients created in this step and $H' \subseteq H$ to denote those that are created but not yet processed by the rounding algorithm. Initially $H := \emptyset$ and $H' := \emptyset$.
%

\smallskip

%
When $\sum_{i \in F'} x'_{i,j} \ge 1/2$ holds for all $j \in D'$,
%$x'$ becomes supporting, 
a cluster centered at a client is formed and possibly rounded, depending on whether or not the client forming the cluster is outlier, and the corresponding parts of the cluster are removed from $F'$, $D'$, and $H'$, respectively.
%
%An intermediate assignment $h$ is defined for such clusters for the analysis of the algorithm. Initially $h := 0$.
%
The cluster forming process repeats until $D' \cup H'$ becomes empty.
%

\smallskip

%
In the second phase the algorithm rounds the remaining clusters centered at the outlier clients using an assignment LP to form an integral multiplicity function.
%
%\smallskip
%
%
In the following we describe the three components of our rounding algorithm in details.
%


%\smallskip

%%%
%%%

\paragraph{Creating the Outlier Clients.}

%
When 
%$x'$ is not supporting and 
$\sum_{i \in F'}x'_{i,j} < 1/2$ for some $j \in J^{(\leftrightarrow)} \cap D'$, the algorithm discards $j$
%, withdraw some of the remaining assignments made for it in $x'$, 
and relocates some of the remaining demand to facilities in $N_{(U,x')}(j)$ to form outlier clients in a way as if the demand were originated from these facilities.
%
%In this step, the demand redistributed forms the outlier clients at the facilities in $N_{(U,x')}(j)$.
%
%We use $H$ to denote the set of outlier clients created through this step and $H' \subseteq H$ to denote those that are not yet processed. Initially $H := \emptyset$ and $H' := \emptyset$.
%

\smallskip

%
%
\begin{wrapfigure}[11]{r}{.3\textwidth}
%\begin{figure*}[h]
\vspace{-4pt}
\enskip
\begin{minipage}{.275\textwidth}
%
\fbox{
\includegraphics[scale=1]{outlier-client-construction}
\hspace{1pt}
}
\caption{%An illustration on the c
Construction of the outlier clients.}
\label{fig-outlier-client-construction}
\end{minipage}
%\end{figure*}
\end{wrapfigure}
%
%In particular, l
Let $r'_j := \min\{ \; \sum_{i \in F'} x'_{i,j} \; , \; \sum_{i \in U} x'_{i,j} \; \}$ be the amount of residue demand of $\vphantom{\text{\Large T}} j$ to be redistributed.
%
For each $w \in N_{(U,x')}(j)$, we create a client $j_w$ at the facility $w$ with demand 
$\vphantom{\text{\Large T}} d_{j_w} := r'_j \cdot x'_{w,j} / \sum_{i \in U}x'_{i,j}$
%$$d_{j_w} := \frac{r'_j}{\sum_{i \in U}x'_{i,j}}\cdot x'_{w,j}$$
and add $j_w$ to both $H$ and $H'$.
%
We set $\vphantom{\text{\Large T}} \alpha_{j_w} := \alpha_j + c_{w,j}$.
%
For each $i \in N_{(F',x')}(j)$, we further set 
%the assignment
$\vphantom{\text{\Large T}} x'_{i,j_w} := d_{j_w} \cdot x'_{i,j} / \sum_{i \in F'}x'_{i,j}.$
%

\smallskip

%
See also Figure~\ref{fig-outlier-client-construction} for an illustration on the construction of $j_w$.
%
%
%\smallskip
%
%
It follows by definition that $d_{j_w} \le x'_{w,j}$, and
% by definition, and 
%Furthermore, 
the residue demand of $j$ is fully redistributed since $\vphantom{\text{\Large T}} \sum_{w \in N_{(U,x')}(j)} d_{j_w} = r'_j$.
%
%$$x'_{i,j_w} := \frac{x'_{i,j}}{\sum_{i \in F'}x'_{i,j}} \cdot d_{j_w}.$$
%
%Furthermore, 
%It follows that
%$\vphantom{\text{\Large T}} \sum_{w \in U} x'_{i,j_w} \le x'_{i,j}$ for all $i \in N_{(F',x')}(j)$, and the capacity constraint for $i$ remains valid.
%
We 
%also 
note that $\vphantom{\text{\Large T}} \sum_{i \in N_{(F',x')}(j)} x'_{i,j_w} = d_{j_w}$, and the demand of $j_w$ is fully-assigned.
% to $N_{(F',x')}(j)$.
%

\smallskip

%
After the outlier client $j_w$ is created for each $w \in N_{(U,x')}(j)$, the algorithm removes $j$ from $D'$ and set $x'_{i,j}$ to be zero for all $i \in F'$.
%
%\mong{**** figure **** properties ****}
%
%\mong{**** note the feasibility for~\ref{cons-P-2} and~\ref{cons-P-3} in~\ref{LP-natural-CFL} ****}
%
%It is easy to see that \mong{****** remains feasible ******}, and $\alpha$ \mong{******* remains valid ********}
%
%\mong{**** define $k \in j$ *** outlier client $k$ is at $j$ ***}
%

%

%

%%%
%%%
%%%

\paragraph{Forming Clusters and Rounding.}

When $\sum_{i \in F'} x'_{i,j} \ge 1/2$ holds for all $j \in D'$,
%$x'$ is supporting, 
the algorithm selects a client $j \in D' \cup H'$ that minimizes $\alpha_j$ to form a cluster.
%
%Intuitively, the selected $j$ has the lowest estimated assignment costs, and any other client in $D' \cup H'$ can afford the relocation cost within the ball $N_{(F',x')}(j)$ centered at $j$.
%
%
%\smallskip
%
%
%\noindent
Depending on the set to which $j$ belongs, the algorithm proceeds differently.
%
%\smallskip
%
\begin{itemize}
\item
%
If $j \in H'$, then a cluster centered at $j$ with satellite facilities $N_{(F',x')}(j)$ is formed.
%
%
We use $B(j) := N_{(F',x')}(j)$ to denote the set of satellite facilities at this moment.
% with satellite facilities $N_{(I \cap F',x')}(j)$.
%
%
%For each $i \in B(j)$ and each $k \in D$, the algorithm increases $x^*_{i,k}$ by $x'_{i,k}$. \mong{** remove? **}
%
%
The algorithm then removes $j$ from $H'$ and $B(j)$ from $F'$.
%
The rounding problem for this cluster is handled later in the second phase of the algorithm.
%

%\smallskip

\item
%
If $j \in D'$, the algorithm further selects a facility $i \in N_{(F',x')}(j)$ with the maximum $u_i$.
%
The algorithm reroutes the assignment and facility values 
%rounds up $y_i$ integrally by absorbing the facility values 
from the facilities in $N_{(F',x')}(j)$ to $i$ as follows.
%
Let 
$\delta_i := \left( \; \frac{1}{2} - y'_i \; \right) / \sum_{k \in N_{(F',x')}(j) \setminus \{i\}}y'_k$
%$$\delta_i := \left( \; \frac{1}{2} - y'_i \; \right) \cdot \frac{1}{\sum_{k \in N_{(F',x')}(j) \setminus \{i\}}y'_k}$$
be the factor 
%on the facility value 
to reroute from
% each facility in 
$N_{(F',x')}(j) \setminus \{i\}$.
%

%\smallskip

%
%The intermediate assignment $h$ is updated as follows.
%
%For each $k \in N_{(D' \cup H', x')}(i)$, we increase $h_{i,k}$ by $x'_{i,k}$.
%
%For each $\ell \in N_{(I \cap F', x')}(j) \setminus \{i\}$ and each $k \in N_{(D'\cup H',x')}(\ell)$, we increase $h_{i,k}$ by $\delta_i \cdot x'_{\ell,k}$.
%, which is 
%Note that this is the amount of assignment facility $i$ absorbs from facility $\ell$.
%
%For each $k \in N_{(D'\cup H',x')}(i)$, also let $h_{i,k} := x'_{i,k}$.
%

%\smallskip

%
For each $\ell \in N_{(F', x')}(j) \setminus \{i\}$, the algorithm 
%then 
scales $y'_\ell$ down by
% a factor of 
$(1-\delta_i)$. 
%
For each $k \in N_{(D'\cup H',x')}(\ell)$, the algorithm further scales $x'_{\ell,k}$ down by $(1-\delta_i)$ and increases 
%the assignment 
$x^*_{i,k}$ by the same amount $x'_{\ell,k}$ has decreased in this step.
% and increases $x^*_{i,k}$ by the same amount $x'_{\ell,k}$ has decreased.

%\smallskip

%
The algorithm increases $x^*_{i,k}$ by $x'_{i,k}$ for each $k \in D'$ and
%
%The algorithm then 
%sets $y^*_i$ to be $1$ and
% $y'_i$ to be zero.
%
%For each $k \in N_{(D' \cup H',x')}(i)$, the algorithm sets 
%$x^*_{i,k}$ to be $x'_{i,k}$ and then 
%$x'_{i,k}$ to be zero.
%
%The algorithm 
then removes $i$ from $F'$.
%

%\smallskip

%
%Intuitively, the algorithm relocates both the facility values and the assignments from facilities in $N_{(F',x')}(j)$ to the facility $i$.
%
%It follows that, for each unit of demand that gets reassigned, say, from facility $\ell$ to facility $i$, the assignment cost is increased by no more than $c_{i,\ell} \le c_{i,k} + c_{\ell,k} \le 2\alpha_j$.
%
%Furthermore,~\ref{cons-P-2} and~\ref{cons-P-3} in~\ref{LP-natural-CFL} remain valid for all satellite facilities in $N_{F',x'}(j)$ after this step.
%
\end{itemize}

%\smallskip

\noindent
%
When the above updates 
%on $(x', y')$ is 
are done, for each $k \in J^{(I)} \cap D'$ with $\sum_{i \in F'} x'_{i,k} < 1/2$, the algorithm removes $k$ from $D'$ and sets $x'_{i,k}$ to be zero for all $i \in F'$.
%
%\mong{**** proceeds to the next iteration ****}
Then the algorithm proceeds to the next iteration until $D' \cup H'$ becomes empty.
%

%

%%
%%

\paragraph{Rounding the Outlier Clusters.}

%
%
\begin{wrapfigure}[14]{r}{.43\textwidth}
%\begin{figure*}[h]
%
\vspace{-14pt}
\enskip\hspace{1pt}
\begin{minipage}{.39\textwidth}
\fbox{\enskip
\begin{minipage}{.96\textwidth}
\vspace{-6pt}
\begin{align}
& \text{min} \enskip \sum_{i \in G} \; y_i \; + \; \sum_{i \in G, \; j \in U} c_{i,j} \cdot x_{i,j} & &  \notag \\[4pt]
& \sum_{i\in G} \; x_{i,j} \; \ge \; d_j, & & \hspace{-42pt} \forall j \in U,  \notag \\[4pt]
& \sum_{j \in U} \; x_{i,j} \; \le \; u_i \cdot y_i, & & \hspace{-42pt} \forall i \in G,  \notag \\[1pt]
& \; y_i \; \le \; 1, & & \hspace{-42pt} \forall i \in G,  \notag \\[5pt]
& \; x_{i,j}, \; y_i \; \ge 0, & & \hspace{-76pt} \forall i \in G, \; j \in U.  \notag
\end{align}
\vspace{-10pt}
\end{minipage}
}
\end{minipage}
%
\caption{LP-(O).}
\label{fig-LP-CFL-outliers-main-content}
%\end{figure*}
\end{wrapfigure}
%
%
When 
%the cluster-forming process ends and 
$D' \cup H'$ becomes empty, the algorithm proceeds to handle the rounding problem for clusters centered at 
%the outlier 
clients in $H$,
%
which is formulated as 
%We formulate this rounding problem as 
an instance of CFL-CFC with facility set $G := \bigcup_{j \in H} B(j)$ and client set $U$ as follows.
%
%
Each 
%client 
$w \in U$ is 
%further 
associated with a demand $d_w$ 
%that needs to be assigned, where $d_w$ is 
defined as 
$$d_w := \sum_{\substack{k \in H, \\[1pt] k \text{ located at } w}} \; \sum_{i \in B(k), \; \ell \in \MC{D}\cup H} \; t'_\ell \cdot x'_{i,\ell},$$
%
%and
where 
%for each $\ell \in \MC{D} \cup H$, 
the scaling factor $t'_\ell$ is defined as
$$t'_\ell \; := \; \frac{1}{\sum_{ k \in I} x^*_{k,\ell} + \sum_{k \in G} x'_{k,\ell} } \cdot \left( \; 1 - \sum_{i \in U} x'_{i,\ell} - r'_\ell \; \right)$$
if $\ell \in \MC{D}$ and $\sum_{k \in I} x^*_{k,\ell} + \sum_{k \in G} x'_{k,\ell} > 0$, and
%$$t'_\ell \; := \; \frac{1}{\sum_{ k \in I} x^*_{k,\ell} + \sum_{k \in G} x'_{k,\ell} } \cdot \left( \; 1 - \sum_{i \in U} x'_{i,\ell} - r'_\ell \; \right) \quad\enskip \text{if \;$\ell \in \MC{D}$\; and \;$\sum_{k \in I} x^*_{k,\ell} > 0$, \enskip and}$$
$t'_\ell := 1$ otherwise.
%
%Note that, it follows 
%from the definition 
%that $t'_\ell \le 2$ for all $\ell \in \MC{D} \cup H$.
%\smallskip
%
Intuitively, $t'_\ell$ is the factor for which the assignments made for $\vphantom{\text{\Large T}} \ell$ 
%in $\bm{x}^*$ 
should be scaled up, and the demand $d_w$ of each $w \in U$ is the total demand of the clusters centered at the outlier clients located at $w$.
% that needs to be assigned.
%

\smallskip

%
The algorithm solves 
%We consider 
the assignment LP for the formulated instance, denoted~\ref{LP-CFL-outliers} and given in Figure~\ref{fig-LP-CFL-outliers-main-content},
%
%\noindent
%The algorithm solves~\ref{LP-CFL-outliers} 
for a basic optimal solution $(\bm{x}'', \bm{y}'')$.
%.
%
%For each $i \in G$, the algorithm 
%and sets $y^*_i := \left\lceil y''_i \right\rceil$ for each $i \in G$.
%

%

%%%
%%%

\paragraph{Final Output.}

%
Define 
%$y^*_i := \begin{cases}
%\; 1, & \text{if $i \in I \setminus F'$,} \\
%\; \left\lceil y''_i \right\rceil, & \text{if $i \in G$, } \\
%\; 0, & \text{otherwise}.
%\end{cases}$.
the multiplicity function $\bm{y}^*$ as follows.
%
Let $y^*_i := 1$ for $i \in I \setminus F'$, \; $\vphantom{\text{\Large T}} y^*_i := \left\lceil y''_i \right\rceil$ for $i \in G$, and $y^*_i := 0$ otherwise.
%
%Provided that the integral multiplicity function $y^*$ is formed, t
The algorithm solves the min-cost assignment problem on $\MC{D}$ and $\MC{F}^* := \{ \hspace{1pt} \vphantom{\text{\Large T}} i \in \MC{F} \; \colon \; y^*_i = 1 \hspace{1pt} \}$ for an optimal integral assignment $\bm{x}^\dagger$,
%
%The algorithm
and reports $(\bm{x}^\dagger, \bm{y}^*)$ as the approximation solution.


%\smallskip

%
%\noindent
%\mong{****} A pseudo-code of this algorithm is provided in Figure~\ref{fig-overview-tight-vc-hc} in Page~\pageref{fig-overview-tight-vc-hc} for further reference.

\smallskip

%It is clear that the algorithm runs in polynomial time.
%
The following theorem, which is proved in Section~\ref{sec-proof-approx-cfl-cfc}, page~\pageref{sec-proof-approx-cfl-cfc}, bounds the cost incurred by $(\bm{x}^\dagger, \bm{y}^*)$ and completes the proof for Theorem~\ref{thm-approx-cfl-ufc}.
%
%\mong{*** pageref ***}

\begin{theorem} \label{theorem-cfl-ufc-approx}
Let $\Psi$ be the input instance of CFL-CFC and $(\bm{x}', \bm{y}')$ be an optimal solution for~\ref{LP-natural-CFL} on $\Psi$.
%
Algorithm $\MC{A}$ computes in polynomial time a feasible integral solution $(x^\dagger, y^*)$ with
$$\psi(x^\dagger, y^*) \le 4\cdot \psi(x',y').$$
\end{theorem}

%

%

%

%\newpage

%

%

%%%
%%%

\section{Proof of Theorem~\ref{thm-MFN-relaxed-separation-oracle}.}
\label{sec-proof-approx-cfl}


It is clear that the first phase of the algorithm runs in polynomial time.
%
Since $\bm{y'} \le \bm{y''}$, the feasible region of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}',\bm{g})$ is contained in that of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$.
%, it follows that,
Hence, any hyperplane separating $(\bm{x}', \bm{y}')$ from $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ also separates $(\bm{x}', \bm{y}')$ from $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}',\bm{g})$.
%
%
%\smallskip
%
%
To complete the proof of Theorem~\ref{thm-MFN-relaxed-separation-oracle}, it suffices to prove the following lemma.
%

\begin{lemma} \label{theorem-cfl-sparse-flow-approx}
Provided that $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ is feasible, our iterative rounding process in the second phase is well-defined and terminates in polynomial time.
Furthermore, the feasible region of the min-cost assignment problem on $\MC{D}$ and $F^*$ is non-empty, and
%
%Furthermore, 
$$\psi(x^*, y^*) \le \max\left\{ \; \frac{3}{2\alpha} \; , \; \frac{7-4\alpha}{(1-\alpha)^2} \; \right\} \times \psi(x',y').$$
\end{lemma}


%
The proof for Lemma~\ref{theorem-cfl-sparse-flow-approx} is outlined as follows. 
%
%To prove this lemma,
First, we define in Section~\ref{subsec-proof-approx-cfl-p1} the intermediate function $\bm{f}'$ which corresponds to the assignments made to facilities in $U$ and in Section~\ref{subsec-proof-approx-cfl-p2} the intermediate assignment $\bm{x}''$ which corresponds to the assignments made by our iterative rounding process to facilities in $I \setminus F'$, respectively.
%algorithm for $\bm{x}'$ and $\bm{y}'$.
%
We prove in the same section that the iterative rounding process is well-defined and runs in polynomial time.
%

\smallskip

%
In Section~\ref{subsec-proof-approx-cfl-feasibility}, page~\pageref{subsec-proof-approx-cfl-feasibility}, we show that, the assignment $\bm{x}'''$ defined jointly from $\bm{f}'$ and $\bm{x}''$, 
%with properly scaled up, 
%say, $\bm{x}^\circ$, meets the statement of this lemma, i.e., $(\bm{x}^\circ, \bm{y}^*)$ 
%to which $\bm{f}''$ corresponds 
together with $\bm{y}^*$ forms a feasible solution for $\Psi$ after properly rounded up.
%
This shows that the feasible region of the min-cost assignment problem on $\MC{D}$ and $F^*$ is nonempty.
%

\smallskip

%
For the approximation guarantee, in Section~\ref{subsec-proof-approx-cfl-approx-guarantee}, page~\pageref{subsec-proof-approx-cfl-approx-guarantee}, we analyze the cost incurred by $\bm{x}'''$ and $\bm{y}^*$, and show that it satisfies the claimed approximation guarantee.
%
Note that, this completes the proof of this lemma since $\bm{x}^*$ is the optimal min-cost assignment between $\MC{D}$ and $\MC{F}^*$.
%

\smallskip

%
%In the following we describe the proof in details.
%
%\mong{**** organization ****}
%
%\mong{**** basic feasible flow *****}
%

\vfill


\begin{figure*}[h]
%\centering
%
\begin{minipage}{\textwidth}
\fbox{
\enskip
\begin{minipage}{.39\textwidth}
\centering
\includegraphics[scale=1]{MFN-construction.pdf}
\vspace{-12pt}
\end{minipage}
\enskip
}
%
\enskip
%
\fbox{
\hspace{-10pt}
\begin{minipage}{.51\textwidth}
\vspace{-2pt}
\begin{subequations}
\begin{align}
& \sum_{i \in \MC{F}, \; p \in P(i,j)} \hspace{-4pt} f_p \; \ge \; r^{(g)}_j, & & \forall j \in \MC{D}, \label{LP_MFN_1} \\[6pt]
& \sum_{p \in P \colon (j^s,i) \in p} \hspace{-6pt} f_p \; \le \; x_{i,j}, & & \hspace{-8pt} \forall i \in \MC{F}, j \in \MC{D}, \label{LP_MFN_2} \\[6pt]
& \sum_{p \in P \colon (i,j^s) \in p} \hspace{-6pt} f_p \; \le \; g_{i,j}, & & \hspace{-8pt} \forall i \in \MC{F}, j \in \MC{D}, \label{LP_MFN_3} \\[6pt]
& \sum_{j \in \MC{D}, \; p \in P(i,j)} \hspace{-10pt} f_p \; \le \; u^{(g)}_i \hspace{-4pt} \cdot y_i, & & \forall i \in \MC{F}, \label{LP_MFN_4} \\[6pt]
& \sum_{p \in P(i,j)} f_p \; \le \; r^{(g)}_j \hspace{-4pt} \cdot y_i, & & \hspace{-8pt} \forall i \in \MC{F}, j \in \MC{D},  \label{LP_MFN_5} \\[6pt]
& \hspace{10pt} f_p \; \ge \; 0, & & \forall p \in P. \label{LP_MFN_6}
\end{align}
\end{subequations}
\vspace{-10pt}
\end{minipage}
\enskip
}
\end{minipage}
\caption{(Restate for further reference) The construction of $\mathbf{MFN}_\Psi(x,y,g)$ and the corresponding LP constraints.}
\label{cons-MFN-g-appendix}
\end{figure*}

%%%
%%%

%\paragraph{The Intermediate Flow $\bm{f}'$ and $\bm{f}''$ for Assignments to $U$.}
\subsection{The Pseudo-Flow Function $\bm{f}'$ for Assignments to $U$}
\label{subsec-proof-approx-cfl-p1}

%
In this section we define the intermediate function $\bm{f}'$ which corresponds to the assignments made to facilities in $U$.\footnote{In the rest of the proof, we will refer $\bm{f}'$ to as a flow function for convenience despite the fact that it uses edges that do not exist in $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$.}
%
Consider the first phase of the algorithm for obtaining the partial assignment $\bm{g}$ and the corresponding feasible flow $\bm{f}$ for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$.
%

\smallskip

%We have the following proposition regarding the clients 
%which is 
%not partially assigned in $\bm{g}$ in this phase, i.e., $j \in \MC{D}$ with $g_{i,j} = 0$ for all $i \in U^{(>)}$.
%The following proposition, which follows from the construction of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ and the feasibility of $\bm{f}$ for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, is crucial in analyzing the clients for which no partial assignments are made in $\bm{g}$.
%


%
Let $D^{(\ell)}$ denote the set of clients that are fully-assigned in $\bm{h}$ and 
%that are 
unreachable from any partially-assigned clients using augmenting paths.
%
Let $D^{(p)} := \MC{D} \setminus D^{(\ell)}$ denote the set of clients that are either partially-assigned or reachable by partially-assigned clients via augmenting paths.
%

\smallskip

%
The following proposition 
%By Proposition~\ref{prop-MFN-flow-client-without-partial-assignment}, it follows
shows that, in the flow $\bm{f}$, none of $j' \in D^{(p)}$ has sent nonzero flow that passes through $j^s$ for any $j \in D^{(\ell)}$, which implies that for any $i \in \MC{F}$ and $j \in D^{(\ell)}$, the arc $(j^s,i)$ carries only flow for commodity $j$.
%

%\medskip

\begin{prop} \label{prop-MFN-flow-client-without-partial-assignment}
For any $j \in D^{(\ell)}$, $i \in \MC{F}$, and any $p \in P$ with $(j^s,i) \in p$, we have $f_p > 0$ implies that $p \in \bigcup_{i' \in \MC{F}} P(i',j)$, that is, $f_p > 0$ only when $p$ is a path for commodity $j$.
% for any $j \in D^{(\ell)}$ and $p \in P(i,j')$ for any $i \in \MC{F}$, $j' \in D^{(p)}$, i.e., 
%in $\bm{f}$, none of $j' \in D^{(p)}$ has sent nonzero flow that passes through $j^s$ for any $j \in D^{(\ell)}$.
%for any $i \in \MC{F}$, $j' \in D^{(p)}$, 
%none of the paths in $P(i,j')$ passes through the node $j^s$.
% for any $j \in D^{(\ell)}$.
%, any , and any $p \in P(i,j)$, we have that $j'^s \notin p$ for all $j' \in D^{(\ell)}$.
%Let $j \in \MC{D}$ be a client with $\sum_{i \in U^{(>)}} g_{i,j} = 0$.
%
%Then, for any $p \in P$ with $(j^s, i) \in p$ for some $i \in \MC{F}$, $\vphantom{\text{\LARGE T}} f_p > 0$ implies that $p \in P(j)$.
\end{prop}

\begin{proof}%[Proof of Proposition~\ref{prop-MFN-flow-client-without-partial-assignment}]
%
Since $j \in D^{(\ell)}$, it follows that $g_{k,j} = 0$ for all $k \in \MC{F}$ by the definition of $\bm{g}$.
%
Hence, $p$ must start from $j^s$ and is thereby a path for commodity $j$.
%only flow that originates from $j^s$ can
%From the construction of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ and the assumption that $\sum_{i \in U^{(>)}} g_{i,j} = 0$, we know that 
%$(i,j^s) \notin \mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ for all $i \in \MC{F}$, which implies that $j^s$ is unreachable from commodities other than $j$.
%
%Hence, any flow carrying nonzero value through the arc $(j^s,i)$ must originate from $j^s$.
\end{proof}


%
\noindent
Define the intermediate flow function $\bm{f}'$ as follows.
%
\begin{itemize}
	\item
		For any $i \in U^{(>)}$ that is tightly-occupied and any $j \in \MC{D}$,
		%
		%\mong{****}
		%(ii) any $i \in U^{(>)}$ that is not tightly-occupied and any $j \in \MC{D}$ that is fully-assigned by $\bm{h}$ in the first phase of the algorithm, 
		%
		we define $$f'_p \; := \; \left( \; 1-\alpha \; \right) \cdot g_{i,j}, \quad \text{where $p = {j^s \rightarrow i \rightarrow i^t \rightarrow j^t}$.}$$
				
	\item
		For any $i \in U^{(>)}$ that is not tightly-occupied, we further consider two cases.
		%
		\begin{itemize}
			\item
			 	For any $j \in D^{(\ell)}$,
			 	%that is fully-assigned by $\bm{h}$ in the first phase of the algorithm and 
				%such that, 
				%there exists no augmenting path connecting a partially-assigned client to $j$, 
				%not connected by an augmenting path starting from a partially-assigned client,
				we define $f'_p \; := \; \left( \; 1-\alpha \; \right) \cdot h_{i,j}, \enskip \text{where $p = {j^s \rightarrow i \rightarrow i^t \rightarrow j^t}$.}$
				
			\item
				For any $j \in D^{(p)}$ and any $p \in P(i,j)$,
				% that is partially-assigned, or, any $j \in \MC{D}$ that is 
				%fully-assigned and 
				%connected by an augmenting path starting from a partially-assigned client,
				%
				we define $f'_p := f_p.$
		\end{itemize}

	\item
		Finally, for 
		%(i) any $i \in U^{(>)}$ that is not tightly-occupied, any $j \in \MC{D}$ that is not fully-assigned by $\bm{h}$ 
		%in the first phase of the algorithm, and any $p \in P(i,j)$, or, 
		%
		%(ii) 
		any $i \in U^{(\le)}$, any $j \in \MC{D}$, and any $p \in P(i,j)$, 
		%
		we define $f'_p := f_p.$
		%
		
%	\item
%		Finally, for any $i \in U^{(le)}$, any $j \in \MC{D}$, and any $p \in P(i,j)$, we define $f'_p := f_p$.
		%
		%Intuitively, we keep the flow sent to $U$ unchanged.
\end{itemize}

%

\smallskip

%
For any $i \in U$, $j \in \MC{D}$, let $\Phi'(i,j)$ denote the set of paths with nonzero flow for commodity $j$ to sink via $i^t$ 
in the flow functions $\bm{f}'$.
% and $\bm{f}''$, respectively.
%
%From the construction of $\bm{f}''$, w
%
The following lemma, which is valid from the construction of $\bm{f}'$, shows that the facilities in $U$ are sparsely-loaded.
% by $\bm{f}''$.
% establishes the feasibility of our rounding algorithm.
%

\begin{lemma} \label{lemma-approx-cfl-U-sparsely-loaded}
For any $i \in U$, we have
%We have
%
%\begin{align*}
%\text{(i)} \quad & \sum_{i \in \MC{F}^*, \; p \in \Phi''(i,j)} f''_p \; \ge \; 1-\alpha  \quad\enskip \text{for all $j \in \MC{D}$, \enskip and} \\[4pt]
$$\sum_{j \in \MC{D}, \; p \in \Phi'(i,j)} f'_p \; \le \; \left( \; 1-\alpha \; \right) \cdot u_i.$$
%\end{align*}
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-approx-cfl-U-sparsely-loaded}]
%
%
%Consider any $i \in \MC{F}^* = U \cup (I \setminus F')$.
%
Depending on the category to which each $i$ belongs, we consider the following three cases separately.
%
\begin{itemize}
	\item
		$i \in U^{(>)}$ is tightly-occupied.
		%
		%For this case, we have $\sum_{j \in \MC{D}} g_{i,j} = u_i$ 
		By the construction rules of $\bm{f}'$ and the definition of $\bm{g}$, we have
		$$\sum_{j \in \MC{D}, \; p \in \Phi'(i,j)} f'_p \;\; = \;\; \sum_{j \in \MC{D}} f'_{j^s \rightarrow i \rightarrow i^t \rightarrow j^t} \; = \; (1-\alpha) \cdot \sum_{j \in \MC{D}} \; g_{i,j} \; = \; (1-\alpha) \cdot \sum_{j \in \MC{D}} \; h_{i,j} \; = \; (1-\alpha) \cdot u_i,$$
		where the last equality follows from the fact that there exists an augmenting path connecting a partially-assigned client to $i$, which implies that $\sum_{j \in \MC{D}} h_{i,j} = u_i$ by the optimality of $\bm{h}$.
		% is tightly-occupied.
		%, it follows that
		%\mong{***}
		
		\smallskip
		
		%
		
	\item
		$i \in U^{(>)}$ is not tightly-occupied.
		%
		%Let $D^{(p)}$ denote the set of partially-assigned clients in $\MC{D}$, $D^{(t)}$
		% := \left\{ j \in \MC{D} \colon \text{ there exists an augmenting path connecting a partially-assigned client to $j$.} \right\}$ 
		%denote the set of fully-assigned clients in $\MC{D}$ that can be reached by partially-assigned clients using augmenting paths,
		%and $D^{(u)} := \MC{D} \setminus ( D^{(p)} \cup D^{(t)} )$.
		%for which there exists an augmenting path connecting a partially-assigned client to it.
		%
		Since $i$ is not tightly-occupied, by the optimality of $\bm{h}$ it follows that, $h_{i,j} = x'_{i,j} / (1-\alpha)$ holds for any $j \in D^{(p)}$.
		%
		By the construction rules of $\bm{f}'$ and
		%
		%Consider the flow sent to sink via $i$ in $\bm{f}$.
		%
		%By 
		Proposition~\ref{prop-MFN-flow-client-without-partial-assignment},
		we have
		%
		\begin{align}
		\sum_{j \in D^{(p)}, p \in \Phi'(i,j)} f'_p \; = \; \sum_{j \in D^{(p)}, \; p \in P(i,j)} f_p \; 
		\le & \; \sum_{\substack{ j \in D^{(p)}, \\[1pt] p \in P(i,j) \text{ with } (j^s,i) \in p }} f_p \;  \notag \\[8pt]
		\le & \; \sum_{j \in D^{(p)}} x'_{i,j} \;\; = \;\; (1-\alpha) \cdot \sum_{j \in D^{(p)}} h_{i,j},
		\label{ieq-cfl-approx-feasibility-1}
		\end{align}
		where in the second last inequality we apply Constraint~(\ref{LP_MFN_2}) due to the fact that $\bm{f}$ is a feasible flow for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$.
		%, 
		%by Constraint~(\ref{LP_MFN_2}) in the constraints of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, 
		%it follows that 
		%
		From the second construction rule of $\bm{f}'$, we have
		\begin{align}
		\sum_{j \in D^{(\ell)}} f'_{j^s \rightarrow i \rightarrow i^t \rightarrow j^t} \; = \; (1-\alpha) \cdot \sum_{j \in D^{(\ell)}} h_{i,j}.
		\label{ieq-cfl-approx-feasibility-2}
		\end{align}
		%
		Combining~(\ref{ieq-cfl-approx-feasibility-1}) and~(\ref{ieq-cfl-approx-feasibility-2}) with the definition of $\Phi'(i,j)$,
		% and applying the construction rules of $\bm{f}'$, 
		we have
		$$\sum_{j \in \MC{D}, \; p \in \Phi'(i,j)} f'_p \; = \; \sum_{j \in D^{(p)}, \; p \in \Phi'(i,j)} f'_p \; + \; \sum_{j \in D^{(\ell)}, \; p \in \Phi'(i,j)} f'_p \;\; \le \;\; (1-\alpha) \cdot u_i,$$
		where the last inequality follows from the feasibility of $\bm{h}$, i.e., $\sum_{j \in \MC{D}} h_{i,j} \le u_i$.
		%, it follows that
		%$$\sum_{j \in \MC{D}} h_{i,j} \; \le \; u_i$$
		%\mong{****}
		%
		%Further applying the feasibility of $\bm{h}$ and the feasibility of $\bm{f}$ for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, we have 
		%\begin{align*}
		%\sum_{j \in D^{(p)} \cup D^{(t)}} x_{i,j} \; \le \; (1-\alpha) \cdot \sum_{j \in D^{(p)} \cup D^{(t)}} h_{i,j} \; \le \; (1-\alpha) \cdot \left( \; u_i \; - \; \sum_{j \in D^{(u)}} h_{i,j} \; \right).
		%\end{align*}
		
		\smallskip
		
		%
				
	\item
		$i \in U^{(\le)}$.
		%
		By the third construction rule of $\bm{f}'$ and the feasibility of $\bm{f}$, we have
		\begin{align*}
		\sum_{j \in \MC{D}, \; p \in \Phi'(i,j)} f'_p \;\; = \; \sum_{j \in \MC{D}, \; p \in P(i,j)} f_p \;\; \le \;\; \sum_{j \in \MC{D}} x'_{i,j} \;\; \le \;\; (1-\alpha)\cdot u_i,
		\end{align*}
		where in the last inequality we apply the definition of $U^{(\le)}$.
		
		\smallskip
		
\end{itemize}
%
This completes the proof of this lemma.
%
\end{proof}

%

%

\begin{figure*}[h]
\centering
\fbox{
\begin{minipage}{.8\textwidth}
\begin{align}
& \text{min} & & \sum_{i \in F'} o_i \cdot y_i \; + \; \sum_{i \in F', \; j \in D'} c_{i,j}\cdot x_{i,j} & &  \label{LP-natural-CFL-modified} \tag*{LP-(M)} \\[3pt]
& \text{s.t.} & & \sum_{i\in F'} \; x_{i,j} \; = \; r'_j, & & \forall j \in D' \label{cons-PM-1} \tag*{(M-1)} \\[3pt]
& & & \sum_{j \in D'} \; x_{i,j} \; \le \; u_i \cdot y_i, & & \forall i \in F' \label{cons-PM-2} \tag*{(M-2)} \\[3pt]
& & & 0 \; \le \; x_{i,j} \; \le \; \frac{2\alpha}{1-\alpha} \cdot r_j \cdot y_i, & & \forall i \in F', j \in D', \label{cons-PM-3} \tag*{(M-3)} \\[5pt]
& & & 0 \; \le \; y_i \; \le \; \frac{1-\alpha}{2}, & & \forall i \in F', \label{cons-PM-4} \tag*{(M-4)}
\end{align}
\vspace{-6pt}
\end{minipage}\quad
}
%
\fbox{
\begin{minipage}{.86\textwidth}
\begin{align}
& \text{max} & & \sum_{j \in D'} r'_j \cdot \lambda_j \; - \; \sum_{i \in F'} \; \frac{1}{2} \cdot (1-\alpha) \cdot \eta_i & &  \label{LP-natural-modified-dual} \tag*{LP-(DM)} \\[6pt]
& \text{s.t.} & & \lambda_j \; \le \; \beta_i \; + \; \Gamma_{i,j} \; + \; c_{i,j}, \enskip & & \forall i \in F', j\in D', & & \label{cons-DM-1} \tag*{(DM-1)} \\[6pt]
& & & u_i \cdot \beta_i \; + \; \frac{2\alpha}{1-\alpha} \cdot \sum_{j \in D'} r_j \cdot \Gamma_{i,j} \; \le \; o_i \; + \; \eta_i, \enskip & & \forall i \in F',  \label{cons-DM-2} \tag*{(DM-2)} \\[1pt]
& & & \lambda_j \in \MBB{R}, \; \beta_i, \; \Gamma_{i,j}, \; \eta_i \; \ge \; 0, & & \forall i \in F', j \in D'. \tag*{(DM-3)}
\end{align}
\vspace{-6pt}
\end{minipage}\quad
}
\caption{(Restate for further reference) The natural LP formulation and its dual LP for our iterative rounding process.}
\label{fig-cfl-natural-modified}
\end{figure*}

%

%%%
%%%

%\paragraph{The Assignment $\bm{x}''$ for Facilities in $I \setminus F'$.}
\subsection{The Assignment $\bm{x}''$ for the Rounded Facilities in $I \setminus F'$}
\label{subsec-proof-approx-cfl-p2}

%
Consider the rounding process in the second phase of the algorithm. With a slight abuse of notations, let $F'$ denote the particular set $F'$ at the moment when the rounding process ends, i.e., $F'$ denotes the set of facilities in $I$ that are not selected to be rounded up by the algorithm.
%
For each $i \in I \setminus F'$,
% that is selected to be rounded by the algorithm, 
let $\vphantom{\text{\Large T}} \Psi^{(i)} = (F'^{(i)}, D'^{(i)},\bm{r'}^{(i)})$ denote the parameter tuple the algorithm maintains in the beginning of the iteration 
%at the moment 
when facility $i$ is selected.
% and about to be rounded.
%
%Also refer to Figure~\ref{fig-cfl-natural-modified} in page~\pageref{fig-cfl-natural-modified} for further reference to the LP formulations used in the iterative rounding process.
%

%

\smallskip

%
Let $\Psi^{(0)} = \left( F'^{(0)}, D'^{(0)}, \bm{r}'^{(0)} \right)$ denote the initial parameter tuple 
the algorithm constructs
in the beginning of the second phase.
%
The following lemma, which shows that the feasible region of~\ref{LP-natural-CFL-modified} on $\Psi^{(0)}$ is nonempty, is proved by verifying the corresponding constraints of~\ref{LP-natural-CFL-modified} combined with the fact that $0 < \alpha \le 1/3$.
%

\begin{lemma} \label{lemma-cfl-approx-iterative-rounding-initial-tuple-feasibility}
The solution $\left(\bm{x}^{\dagger(0)}, \bm{y}^{\dagger(0)}\right)$ defined by
\begin{align*}
x^{\dagger(0)}_{i,j} := \sum_{p \in P(i,j)} f_p \quad \text{for all $i \in I, j \in \MC{D}$} \quad\enskip \text{and} \quad y^{\dagger(0)}_i := \frac{1-\alpha}{2\alpha} \cdot y'_i \quad \text{for all $i \in I$}
\end{align*}
is feasible for~\ref{LP-natural-CFL-modified} on the initial parameter tuple $\Psi^{(0)}$.
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cfl-approx-iterative-rounding-initial-tuple-feasibility}]
%
We prove this lemma by verifying the constraints of~\ref{LP-natural-CFL-modified} separately.
%
\begin{itemize}
	\item
		For Constraint~\ref{cons-PM-1}, consider any $j \in D'^{(0)}$.
		%
		Since $F^{(0)} = I$ by definition, further applying the definition of $r'^{(0)}_j$, we have
		\begin{align*}
		\sum_{i \in F'^{(0)}} x^{\dagger(0)}_{i,j} \; = \; \sum_{i \in I, \; p \in P(i,j)} f_p \; = \; r'^{(0)}_j.
		\end{align*}
		
	\item
		For Constraint~\ref{cons-PM-2}, consider any $i \in F'^{(0)}$.
		%
		By the definition of $D'^{(0)}$ and Constraint~(\ref{LP_MFN_4}) from the fact that $\bm{f}$ is a feasible flow for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, it follows that
		\begin{align*}
		\sum_{j \in D'^{(0)}} x^{\dagger(0)}_{i,j} \; = \; \sum_{j \in D'^{(0)}, \; p \in P(i,j)} f_p \; \le \; \sum_{j \in \MC{D}, \; p \in P(i,j)} f_p \; \le \; u^{(g)}_i \cdot y'_i.
		\end{align*}
		%where in the last inequality we apply Constraint~(\ref{LP_MFN_4}) of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$.
		%
		
		Since $F'^{(0)} := I$ by definition and $i \in F'^{(0)}$, it follows that $\sum_{j \in \MC{D}} g_{i,j} = 0$ by the way $\bm{g}$ is defined and therefore $\vphantom{\text{\Large T}} u^{(g)}_i = u_i$.
		%
		Moreover, observe that $(1-\alpha)/(2\alpha)$ is strictly decreasing for $\alpha > 0$.
		%
		Since $\vphantom{\text{\Large T}} 0 < \alpha \le 1/3$, it follows that $$y^{\dagger(0)}_i \; := \; \frac{1-\alpha}{2\alpha} \cdot y'_i \; \ge \; \frac{ 1- 1/3}{2/3} \cdot y'_i \; = \; y'_i.$$
		%
		Combining the above, we obtain
		$$\sum_{j \in D'^{(0)}} x^{\dagger(0)}_{i,j} \; \le \; u_i \cdot y^{\dagger(0)}_i.$$
		
	\item
		For Constraint~\ref{cons-PM-3}, consider any $i \in F'^{(0)}$ and any $j \in D'^{(0)}$.
		%
		Apply Constraint~(\ref{LP_MFN_5}) by the fact that $\bm{f}$ is a feasible flow for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ and the definition of $y^{\dagger(0)}_i$ and $r_j$, we have
		\begin{align*}
		x^{\dagger(0)}_{i,j} \; = \; \sum_{p \in P(i,j)} f_p \; \le \; r^{(g)}_j \cdot y'_i \; = \; \frac{2\alpha}{1-\alpha} \cdot r_j \cdot y^{\dagger(0)}_i.
		\end{align*}
		
	\item
		For Constraint~\ref{cons-PM-4}, consider any $i \in F'^{(0)}$.
		%
		By the definition of $y^{\dagger(0)}_i$ and the fact that $F'^{(0)} = I$, which implies that $y'_i < \alpha$, it follows that
		$$y^{\dagger(0)}_i \; = \; \frac{1-\alpha}{2\alpha} \cdot y'_i \; \le \; \frac{1-\alpha}{2}.$$
\end{itemize}
%
This proves the lemma.
%
\end{proof}

%

\smallskip

%
Consider any $i \in I \setminus F'$ and the iteration when $i$ is processed.
%
Suppose for the moment that the feasible region of~\ref{LP-natural-CFL-modified} on $\Psi^{(i)}$ is nonempty.
%
Let
%
$\left(\bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right)$
% and $(\bm{\lambda}^{\dagger(i)}, \bm{\beta}^{\dagger(i)}, \bm{\Gamma}^{\dagger(i)}, \bm{\eta}^{\dagger(i)})$ 
denote the optimal solution the algorithm computes for~\ref{LP-natural-CFL-modified}
% and~\ref{LP-natural-modified-dual} 
on $\Psi^{(i)}$.
% during the iteration when $i$ is processed.
%
The following lemma shows that the scaled-down operation in each iteration is well-defined.
%

\begin{lemma} \label{lemma-cfl-approx-iterative-rounding-well-defined}
For any $i \in I \setminus F'$ with $y^{\dagger(i)}_i < (1-\alpha)/2$, the following holds.
%
\begin{itemize}
	\item
		For any $j \in D'^{(i)}$, we have $$0 \; \le \; \delta^{(i)}_j \; \le \; \sum_{\ell \in F'^{(i)} \setminus \{i\}} x^{\dagger(i)}_{\ell,j}.$$
		
	\item
		For any $k \in F'^{(i)} \setminus \{i\}$, we have $\sigma^{(i)}_k \; \le \; 1$.
\end{itemize}
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cfl-approx-iterative-rounding-well-defined}]
%
Consider any $j \in D'^{(i)}$.
%
Since $\left( \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right)$ is feasible for~\ref{LP-natural-CFL-modified} on $\Psi^{(i)}$, 
we have $y^{\dagger(i)}_i \le (1-\alpha)/2$, which implies that $(1-\alpha) / (2 \cdot y^{\dagger(i)}_i) \ge 1$.
%
Hence $$\delta^{(i)}_j \; := \; \left( \; \frac{1-\alpha}{2\cdot y^{\dagger(i)}_i} -1 \; \right) \cdot x^{\dagger(i)}_{i,j} \; \ge \; 0.$$
%
On the other hand, applying Constraint~\ref{cons-PM-3} of~\ref{LP-natural-CFL-modified}, it follows that
$$\delta^{(i)}_j \; := \; \frac{1-\alpha}{2\cdot y^{\dagger(i)}_i} \cdot x^{\dagger(i)}_{i,j} - x^{\dagger(i)}_{i,j} \; \le \; \alpha\cdot r_j - x^{\dagger(i)}_{i,j} \; < \; r'^{(i)}_j - x^{\dagger(i)}_{i,j} \; = \; \sum_{\ell \in F'^{(i)} \setminus \{i\}} x^{\dagger(i)}_{\ell,j},$$
where in the second last inequality we apply the fact that $r'^{(i)}_j > \alpha \cdot r_j$ holds for all $j \in D'^{(i)}$ by the design of the algorithm, and in the last equality we apply Constraint~\ref{cons-PM-1} of~\ref{LP-natural-CFL-modified}.
%
This proves the first part of this lemma.
%

\smallskip

%
Consider any $k \in F'^{(i)} \setminus \{i\}$.
%
By the conclusion of the first part, for any $j \in D'^{(i)}$, we have 
$$\sigma^{(i)}_{k,j} \; := \; \frac{\delta^{(i)}_j}{\sum_{ \ell \in F'^{(i)} \setminus \{i\}} x^{\dagger(i)}_{\ell,j}} \cdot x^{\dagger(i)}_{k,j} \; \le \; x^{\dagger(i)}_{k,j},$$
which implies that
$$\sigma^{(i)}_k \; := \; \frac{1}{\sum_{\ell \in D'^{(i)}} x^{\dagger(i)}_{k,\ell}} \cdot \sum_{j \in D'^{(i)}} \sigma^{(i)}_{k,j} \; \le \; 1.$$
%
\end{proof}

%

%\smallskip

%
Let $\Psi''^{(i)} = (F''^{(i)}, D''^{(i)},\bm{r''}^{(i)})$ denote the updated parameter tuple the algorithm maintains at the end of the iteration $i$.
%
Note that, by the algorithm design, $F''^{(i)} := F'^{(i)} \setminus \{i\}$.
%
The following lemma shows that the feasible region of~\ref{LP-natural-CFL-modified} on the updated tuple $\Psi''^{(i)}$ is nonempty and establishes the feasibility of 
%the parameter tuples considered in 
our iterative rounding process.

\begin{lemma} \label{lemma-cfl-approx-iterative-rounding-successive-tuple-feasibility}
For any $i \in I \setminus F'$, 
%we have $\sigma^{(i)}_k \le 1$ for all $k \in F''^{(i)}$.
%
%Furthermore, 
the solution $\left(\bm{x'}^{\dagger(i)}, \bm{y'}^{\dagger(i)}\right)$
%, which is 
defined by
\begin{align*}
x'^{\dagger(i)}_{k,j} := & \left(1-\sigma^{(i)}_k\right) \cdot x^{\dagger(i)}_{k,j} \quad \text{for any $k \in F''^{(i)}, j \in D''^{(i)}$}, \\[6pt]
y'^{\dagger(i)}_k := & \left(1-\sigma^{(i)}_k\right) \cdot y^{\dagger(i)}_k \quad \text{for any $k \in F''^{(i)}$},
\end{align*}
is feasible for~\ref{LP-natural-CFL-modified} on the updated parameter tuple $\Psi''^{(i)}$.
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cfl-approx-iterative-rounding-successive-tuple-feasibility}]
%
%
We prove this lemma by verifying the corresponding constraints in~\ref{LP-natural-CFL-modified}.
%
First, by Lemma~\ref{lemma-cfl-approx-iterative-rounding-well-defined}, 
%it follows that 
$1-\sigma^{(i)}_k \ge 0$ holds for all $k \in F''^{(i)}$, and $\left(\bm{x'}^{\dagger(i)}, \bm{y'}^{\dagger(i)}\right)$ is thereby nonnegative.
% constraints.
%
%
\begin{itemize}
	\item
		For Constraint~\ref{cons-PM-1}, consider any $j \in D''^{(i)}$.
		%
		By the definition of ${x'}^{\dagger(i)}_{k,j}$ for any $k \in F''^{(i)}$ and the definition of $r''^{(i)}_j$, we have
		$$\sum_{k \in F''^{(i)}} {x'}^{\dagger(i)}_{k,j} \; = \; \sum_{k \in F''^{(i)}} \left( 1-\sigma^{(i)}_k \right) \cdot x^{\dagger(i)}_{k,j} \; = \; r''^{(i)}_j.$$
		
	\item
		For Constraint~\ref{cons-PM-2} and~\ref{cons-PM-3}, consider any $k \in F''^{(i)}$ and any $j \in D''^{(i)}$.
		%
		Since $\left( \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right)$ is feasible for~\ref{LP-natural-CFL-modified} on $\Psi^{(i)}$ and since $F'^{(i)} \supseteq F''^{(i)}$, $D'^{(i)} \supseteq D''^{(i)}$, we have
		$$\sum_{j \in D''^{(i)}} x^{\dagger(i)}_{k,j} \; \le \; \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{k,j} \; \le \; u_k \cdot y^{\dagger(i)}_k \quad\enskip \text{and} \quad\enskip x^{\dagger(i)}_{k,j} \; \le \; \frac{2\alpha}{1-\alpha} \cdot r_j \cdot y^{\dagger(i)}_k.$$
		%
		Multiplying both inequalities by $\left(1-\sigma^{(i)}_k\right)$ and applying the definition of $\left(\bm{x'}^{\dagger(i)}, \bm{y'}^{\dagger(i)}\right)$, we obtain
		$$\sum_{j \in D''^{(i)}} {x'}^{\dagger(i)}_{k,j} \; \le \; u_k \cdot {y'}^{\dagger(i)}_k \quad\enskip \text{and} \quad\enskip {x'}^{\dagger(i)}_{k,j} \; \le \; \frac{2\alpha}{1-\alpha} \cdot r_j \cdot {y'}^{\dagger(i)}_k.$$
		
	\item
		For Constraint~\ref{cons-PM-4}, consider any $k \in F''^{(i)}$.
		%
		Applying the definition of ${y'}^{\dagger(i)}_k$ and the fact that $\left( \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right)$ is feasible for~\ref{LP-natural-CFL-modified}, we have
		$${y'}^{\dagger(i)}_k \; = \; \left( 1-\sigma^{(i)}_k \right) \cdot y^{\dagger(i)}_k \; \le \; y^{\dagger(i)}_k \; \le \; \frac{1-\alpha}{2}.$$
\end{itemize}
%
This proves the lemma.
%
\end{proof}

%

\medskip

%
Lemma~\ref{lemma-cfl-approx-iterative-rounding-initial-tuple-feasibility}, Lemma~\ref{lemma-cfl-approx-iterative-rounding-well-defined}, and Lemma~\ref{lemma-cfl-approx-iterative-rounding-successive-tuple-feasibility} jointly establish that our iterative rounding process is well-defined.
%
Since $F'^{(0)} := I$ and since exactly one facility is selected to be removed from $F'$ in each iteration, it follows that the process repeats for at most $|I|$ iterations before the set $D'$ becomes empty.
%
Therefore, our iterative rounding process terminates in polynomial time.
%

\smallskip

%
For any $i \in I \setminus F'$ and $j \in \MC{D}$, define the assignment $\bm{x}''$ as 
$$x''_{i,j} \; := \; \sum_{k \in F'^{(i)}} \sigma^{(i)}_k \cdot x^{\dagger(i)}_{k,j},$$ where in the above we extend the original definition of $\bm{\sigma}^{(i)}$ and define $\sigma^{(i)}_i := 1$ for notational brevity.
%
Recall that, for any $j \in \MC{D}$, the initial residue demand of $j$ is defined as 
$$r'^{(0)}_j \; = \; \sum_{i \in I, \; p \in P'(i,j)} f_p = \sum_{i \in I, \; p \in P(i,j)} f_p.$$
%
The following lemma, which is valid from the algorithm design, shows that the residue demand of each $j$ is mostly assigned by our rounding process to facilities in $I \setminus F'$.
%\mong{*****}

\begin{lemma} \label{lemma-approx-cfl-I-mostly-assigned}
For any $j \in \MC{D}$, we have
$$\sum_{i \in I \setminus F'} x''_{i,j} \; \ge \; r'^{(0)}_j - \alpha \cdot r_j.$$
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-approx-cfl-I-mostly-assigned}]
%
By Lemma~\ref{lemma-cfl-approx-iterative-rounding-initial-tuple-feasibility}, Lemma~\ref{lemma-cfl-approx-iterative-rounding-successive-tuple-feasibility}, the definition of $r'^{(i)}_j$ and the definition of $x''_{i,j}$ for each $i \in I \setminus F'$, it follows that, each unit of assignment of any $j \in \MC{D}$ that is scaled down during the iteration for $i \in I \setminus F'$ is included in $\sum_{i \in I \setminus F'} x''_{i,j}$.
%
%each unit of assignment of $j$ that is lost during the rounding process due to scaled down is included in $\sum_{i \in I \setminus F'} x''_{i,j}$.
%
Since the algorithm removes a client $j$ from $D'$ only when its residue demand $r'_j$ is no greater than $\alpha \cdot r_j$ and since the initial residue demand of $j$ is $r'^{(0)}_j$, it follows that
$\sum_{i \in I \setminus F'} x''_{i,j} \; \ge \; r'^{(0)}_j - \alpha \cdot r_j$.
%
\end{proof}

%
The following lemma shows that the facilities in $I \setminus F'$ is also sparsely-loaded under the assignments given in $\bm{x}''$.
%

\begin{lemma} \label{lemma-approx-cfl-I-sparsely-loaded}
For any $i \in I \setminus F'$, we have
%
\begin{align*}
& \sum_{j \in \MC{D}} x''_{i,j} \;\; = \;\; \frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} \cdot \sum_{\ell \in D'^{(i)}} x^{\dagger(i)}_{i,\ell} \;\; \le \;\; \frac{1-\alpha}{2} \cdot u_i.
\end{align*}
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-approx-cfl-I-sparsely-loaded}]
%
By the definition of $\bm{x}''$ and the definition of $\sigma^{(i)}_k$ for all $k \in F'^{(i)}$,
we have
\begin{align}
\sum_{j \in \MC{D}} x''_{i,j} \;\; 
= & \;\; \sum_{j \in D'^{(i)}, \; k \in F'^{(i)}} \sigma^{(i)}_k \cdot x^{\dagger(i)}_{k,j} \;\;  \notag \\[8pt]
= & \;\; \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{i,j} \; + \; \sum_{k \in F'^{(i)} \setminus \{i\}} \frac{1}{\sum_{\ell \in D'^{(i)}} x^{\dagger(i)}_{k,\ell}} \sum_{\ell \in D'^{(i)}} \sigma^{(i)}_{k,\ell} \cdot \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{k,j} \;\; \notag \\[8pt]
= & \;\; \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{i,j} \; + \; \sum_{k \in F'^{(i)} \setminus \{i\}, \; \ell \in D'^{(i)}} \sigma^{(i)}_{k,\ell} .
\label{ieq-approx-cfl-I-sparse-loaded-1}
\end{align}
%
Consider the second item in~(\ref{ieq-approx-cfl-I-sparse-loaded-1}).
%
Further applying the definition of $\sigma^{(i)}_{k,\ell}$ for all $k \in F'^{(i)} \setminus\{i\}$, $\ell \in D'^{(i)}$ and the definition of $\delta^{(i)}_\ell$,
% on the second item, 
we obtain
\begin{align}
%& \sum_{k \in F'^{(i)} \setminus \{i\}} \frac{1}{\sum_{\ell \in D'^{(i)}} x^{\dagger(i)}_{k,\ell}} \sum_{\ell \in D'^{(i)}} \sigma^{(i)}_{k,\ell} \cdot \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{k,j} \;\; = \;\; 
\sum_{k \in F'^{(i)} \setminus \{i\}, \; \ell \in D'^{(i)}} \sigma^{(i)}_{k,\ell} \;\; = & \;\; \sum_{k \in F'^{(i)} \setminus \{i\}, \ell \in D'^{(i)}} \frac{x^{\dagger(i)}_{k,\ell}}{\sum_{p \in F'^{(i)} \setminus \{i\}} x^{\dagger(i)}_{p,\ell}} \cdot \delta^{(i)}_\ell \;\;  \notag \\[8pt]
= & \;\; \sum_{\ell \in D'^{(i)}} \delta^{(i)}_\ell \;\; = \;\; \sum_{\ell \in D'^{(i)}} \left( \; \frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} -1 \; \right) \cdot x^{\dagger(i)}_{i,\ell}.
\label{ieq-approx-cfl-I-sparse-loaded-2}
\end{align}
%

\smallskip

%
%\noindent
Combining~(\ref{ieq-approx-cfl-I-sparse-loaded-1}) and~(\ref{ieq-approx-cfl-I-sparse-loaded-2}), and applying applying~\ref{cons-PM-2} in~\ref{LP-natural-CFL-modified} by the fact that $\left(\bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)}\right)$ is a feasible solution for~\ref{LP-natural-CFL-modified} on $\Psi^{(i)}$, we have
\begin{align*}
\sum_{j \in \MC{D}} x''_{i,j} \;\; = \;\; \frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} \cdot \sum_{\ell \in D'^{(i)}} x^{\dagger(i)}_{i,\ell} \;\; \le \;\; \frac{1-\alpha}{2} \cdot u_i.
\end{align*}
%
This completes the proof of this lemma.
\end{proof}

%

%

%

%%%%
%%%%
%%%%

%\paragraph{The Assignment $\bm{x}'''$ and the Feasibility.}
\subsection{The Overall Assignment $\bm{x}'''$ and the Feasibility}
\label{subsec-proof-approx-cfl-feasibility}

%
Define the assignment $\bm{x}'''$ as follows. 
%
For any $i \in \MC{F}^*$ and $j \in \MC{D}$, let 
$$x'''_{i,j} \; := \; 
\begin{cases}
\;\; \sum_{p \in \Phi'(i,j)} f'_p, & \text{if $i \in U$,} \\[4pt]
\;\; x''_{i,j}, & \text{if $i \in I \setminus F'$.}
\end{cases}$$
%
%\smallskip
%
%
%\noindent
%\mong{******}
%
The following lemma establishes the feasibility of our rounding algorithm.
%

\begin{lemma} \label{lemma-approx-cfl-flow-feasibility}
For any $j \in \MC{D}$, we have
%
\begin{align*}
\sum_{i \in \MC{F}^*} x'''_{i,j} \; \ge \; 1-\alpha.
\end{align*}
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-approx-cfl-flow-feasibility}]
%
By the definition of $\bm{x}'''$, to prove this lemma, it suffices to prove that
$$\sum_{i \in \MC{F}^*} x'''_{i,j} \;\; 
= \;\; \sum_{i \in U, \; p \in \Phi'(i,j)} f'_p \; + \; \sum_{i \in I \setminus F'} x''_{i,j} \;\; \ge \;\; 1-\alpha.$$
%
%
%\smallskip
%
%
Depending on the category to which $j$ belongs in the first phase of the algorithm, we consider the following two cases.
%
\begin{itemize}
	\item
		$j \in D^{(\ell)}$, i.e., 
		$j$ is fully-assigned by $\bm{h}$ to facilities in $U^{(>)}$ 
		%in the first phase of the algorithm 
		and unreachable from any partially-assigned client via augmenting paths.
		% starting from a partially-assigned client.
		
		%
		Since $j$ is fully-assigned by $\bm{h}$ to $U^{(>)}$, we have
		$$\sum_{i \in U^{(>)}} h_{i,j} \; = \; 1.$$
		%
		Furthermore, since $j$ is unreachable from any partially-assigned client via augmenting paths, 
		it follows that for any $i \in U^{(>)}$, $h_{i,j} > 0$ implies that $i$ is not tightly-occupied.
		%
		
		Hence, by the construction rules of $\bm{f}'$, it follows that 
		$$\sum_{i \in U, \; p \in \Phi'(i,j)} f'_p \; + \; \sum_{i \in I \setminus F'} x''_{i,j} \; \ge \; \sum_{i \in U^{(>)}} f'_{j^s \rightarrow i \rightarrow i^t \rightarrow j^t} \; = \; (1-\alpha) \sum_{i \in U^{(>)}} h_{i,j} \; = \; 1-\alpha.$$
		%
		
		\smallskip
		
	\item
		For the other case, $j \in D^{(p)}$, i.e., $j$ is either partially-assigned or reachable from partially-assigned clients via augmenting paths,
		%$j$ is fully-assigned but there exists an augmenting path connecting a partially-assigned client to $j$,
		%by $\bm{h}$ to $U^{(>)}$.
		
since $\bm{f}$ is feasible for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, it follows from constraint~(\ref{LP_MFN_1}) and the definition of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ that 
\begin{equation}
\sum_{i \in U, \; p \in P(i,j)} f_p \; + \; \sum_{i \in I, \; p \in P(i,j)} f_p \; = \; r_j \; := \; 1 \; - \; \sum_{i \in U} \; g_{i,j}.
\label{ieq-approx-cfl-flow-feasibility}
\end{equation}
%

\smallskip

%
%By Proposition~\ref{prop-MFN-flow-client-without-partial-assignment}, in $\bm{f}$ no flow of commodity $j$ is sent to\mong{****}
Consider the two items in the LHS of~(\ref{ieq-approx-cfl-flow-feasibility}) separately.
%
By the construction of $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$, no flow is sent in $\bm{f}$ to sink via tightly-occupied facilities in $U^{(>)}$ since $(i,i^t)$ has zero capacity for each tightly-occupied $i \in U^{(>)}$.
%

\smallskip

Hence, by the second and third construction rules of $\bm{f}'$, we know that the flow sent in the first item, $\sum_{i \in U, \; p \in P(i,j)} f_p$, is included entirely in $\bm{f}'$, i.e., 
\begin{equation}
\sum_{\substack{i \in U^{(>)}, \\[1pt] i \text{ not tightly-occupied}}} \sum_{p \in \Phi'(i,j)} f'_p \; + \; \sum_{i \in U^{(\le)}, \; p \in \Phi'(i,j)} f'_p \;\; = \;\; \sum_{i \in U, \; p \in P(i,j)} f_p.
\label{ieq-approx-cfl-flow-feasibility-2}
\end{equation}
%

\smallskip

%
For the second item in the LHS of~(\ref{ieq-approx-cfl-flow-feasibility}), $\sum_{i \in I, \; p \in P(i,j)} f_p$, consider the rounding process for the clusters in $I \setminus F'$.
%
Applying Lemma~\ref{lemma-approx-cfl-I-mostly-assigned} and the definition of $\vphantom{\text{\Large T}} r'^{(0)}_j$, we have
\begin{equation}
\sum_{i \in I \setminus F'} x''_{i,j} \;\; \ge \;\; r'^{(0)}_j - \alpha \cdot r_j \;\; = \;\; \sum_{i \in I, \; p \in P(i,j)} f_p - \alpha \cdot r_j.
\label{ieq-approx-cfl-flow-feasibility-3}
\end{equation}
%

\smallskip

%
By the first construction rule of $\bm{f}'$, we know that exactly $(1-\alpha) \cdot \sum_{i \in U} g_{i,j}$ units of flow
%, out of $\sum_{i \in U} g_{i,j}$, 
is included in $\bm{f}'$.
%
Therefore, 
\begin{equation}
\sum_{\substack{i \in U^{(>)}, \\[1pt] i \text{ tightly-occupied}}} \sum_{p \in \Phi'(i,j)} f'_p \;\; = \;\; (1-\alpha) \cdot \sum_{\substack{i \in U^{(>)}, \\[1pt] i \text{ tightly-occupied} }} g_{i,j} \;\; = \;\; (1-\alpha) \cdot \sum_{i \in U} g_{i,j}.
\label{ieq-approx-cfl-flow-feasibility-4}
\end{equation}

%
Combining the definition of $U$ with~(\ref{ieq-approx-cfl-flow-feasibility-2}), we obtain
\begin{align*}
\sum_{i \in U, \; p \in \Phi'(i,j)} f'_p \; + \; \sum_{i \in I \setminus F'} x''_{i,j} \;\; 
= & \sum_{\substack{i \in U^{(>)}, \\[1pt] i \text{ tightly-occupied}}} \sum_{p \in \Phi'(i,j)} f'_p \; + \; \sum_{i \in U, \; p \in P(i,j)} f_p \; + \; \sum_{i \in I \setminus F'} x''_{i,j}  \\[8pt]
\ge & \;\;\; (1-\alpha) \cdot \sum_{i \in U} g_{i,j} \; + \; (1-\alpha) \cdot r_j \;\; = \;\; 1-\alpha,
\end{align*}
where in the second inequality we apply~(\ref{ieq-approx-cfl-flow-feasibility-4}),~(\ref{ieq-approx-cfl-flow-feasibility-3}), and then~(\ref{ieq-approx-cfl-flow-feasibility}), respectively, and in the last equality we apply the definition of $r_j$.
%

%
%Combining~(\ref{ieq-approx-cfl-flow-feasibility-2}),~(\ref{ieq-approx-cfl-flow-feasibility-3}) and applying the definition of $U$, \mong{****} with~(\ref{ieq-approx-cfl-flow-feasibility}) and~(\ref{ieq-approx-cfl-flow-feasibility-4}), we obtain
%\begin{align*}
%\sum_{i \in U, \; p \in \Phi'(i,j)} f'_p \; + \; \sum_{i \in I \setminus F'} x''_{i,j} \;\; \ge \;\; (1-\alpha) \cdot r_j \; + \; (1-\alpha) \cdot (1-r_j) \;\; = \;\; 1-\alpha.
%\end{align*}

%
%Since the construction of $\bm{f}''$ preserves the amount of flow $\bm{f}'$ sends for $j$, it follows that
%$$\sum_{i \in F^*, \; p \in \Phi'(i,j)} f''_p \; = \; \sum_{i \in F^*, \; p \in \Phi(i,j)} f'_p \; \ge \; 1-\alpha.$$
%
\end{itemize}
%
This completes the proof of this lemma.
%
\end{proof}

%

%
%For any $i \in I \setminus F'$ and any $j \in D'^{(i)}$, define the aggregated assignment
%$$h''_{i,j} \; := \; \sum_{p \in P(i,j)} f^{(i)}_p \; + \; \sum_{k \in F'^{(i)} \setminus \{i\}} \; \sum_{p \in \Phi(i,k,j)} f'_p.$$
%
%For any $i \in U$, $j \in \MC{D}$, we keep the original flow assignment and define $h''_{i,j} \; := \; \sum_{p \in P(i,j)} f_p$.
%

%\smallskip

%
%For any $i \in \MC{F}^*$ and any $j \in \MC{D}$, define the assignment
%$$x''_{i,j} \; := \; 
%\begin{cases}
%\; g_{i,j}, & \text{if $i \in U^{(>)}$ is tightly-occupied,} \\[4pt]
%\; h_{i,j}, & \text{if $i \in U^{(>)}$ is not tightly-occupied and $j$ is fully-assigned,} \\[4pt]
%\; \frac{1}{1-\alpha} \cdot \sum_{p \in P(i,j)} f_p, & \text{if $i \in U^{(>)}$ is not tightly-occupied and $j$ is partially-assigned,} \\
%\; \frac{1}{1-\alpha} \cdot \sum_{p \in P(i,j)} f_p, & \text{if $i \in U^{(\le)}$,} \\
%\; \frac{1}{1-\alpha} \cdot \sum_{p \in \Phi(i,j)} f'_p, \enskip & \text{otherwise.} \\[-10pt]
%\phantom{tt}
%\end{cases}$$
%



%
%Combining with
It follows directly from Lemma~\ref{lemma-approx-cfl-flow-feasibility}, Lemma~\ref{lemma-approx-cfl-U-sparsely-loaded}, and Lemma~\ref{lemma-approx-cfl-I-sparsely-loaded} that $\left( \; \bm{x}'''/(1-\alpha), \; \bm{y}^* \; \right)$ is a feasible solution for $\Psi$, and 
the feasible region of the min-cost assignment problem on $\MC{D}$ and $\MC{F}^*$ is thereby nonempty.
%
We have the following corollary.
%we have the following corollary, which implies that .
%
%\mong{*** for any $\bm{f}$ satisfying Lemma 8 ***}
%

\begin{corollary} \label{lemma-approx-cfl-feasibility}
$\left( \; \frac{1}{1-\alpha} \cdot \bm{x}''', \; \bm{y}^* \; \right)$ is a feasible solution for $\Psi$.
%, where for any $i \in \MC{F}^*$ and $j \in \MC{D}$, the assignment $\bm{x}''$ is defined as $$x''_{i,j} \; := \; \frac{1}{1-\alpha} \cdot \sum_{p \in \Phi''(i,j)} f''_p.$$
\end{corollary}

%\begin{proof}[Proof of Corollary~\ref{lemma-approx-cfl-feasibility}]
%
%\mong{*******}
%
%\end{proof}

%

%%%
%%%
%%%

%\paragraph{Approximation Guarantee.}
\subsection{Approximation Guarantee}
\label{subsec-proof-approx-cfl-approx-guarantee}

%
In the following we bound the cost incurred by facilities in $\MC{F}^*$ and the assignment cost incurred by $\bm{x}''' / (1-\alpha)$.
%
Note that this establishes an upper-bound on the cost incurred by $(\bm{x}^*, \bm{y}^*)$ since $\bm{x}^*$ is the optimal solution of the min-cost problem between $\MC{F}^*$ and $\MC{D}$.
%

\smallskip

%
For any $i \in U, j \in \MC{D}$ and any $p \in \Phi'(i,j) \cup P(i,j)$, define the absolute length of path $p$ as 
%any $p \in \Phi'(i,j) \cup \Phi''(i,j)$ as
%\mong{*** $\Phi''(i,j)$ only? ***}
$$|p| \; := \; \sum_{\substack{i' \in \MC{F}, \; j' \in \MC{D}, \\[2pt] (j'^s,i') \in p}} c_{i',j'} \; + \; \sum_{\substack{ i' \in \MC{F}, \; j' \in \MC{D}, \\[2pt] (i',j'^s) \in p}} c_{i',j'}. $$
%
By the definition of $\bm{x}'''$ and the triangle inequality $c_{i,j} \le |p|$ which holds for any $i \in U, j \in \MC{D}$, and any $p \in \Phi'(i,j)$, we have
\begin{align}
\sum_{i \in \MC{F}^*, \; j \in \MC{D}} c_{i,j} \cdot x'''_{i,j} \;\; 
& = \;\; \sum_{i \in U, \; j \in \MC{D}} \; \sum_{p \in \Phi'(i,j)} c_{i,j} \cdot f'_p \; + \; \sum_{i \in I \setminus F', \; j \in \MC{D}} c_{i,j} \cdot x''_{i,j}  \notag \\[8pt]
& \le \;\; \sum_{i \in U,\; j \in \MC{D}} \; \sum_{p \in \Phi'(i,j)} |p| \cdot f'_p \; + \; \sum_{i \in I \setminus F', \; j \in \MC{D}} c_{i,j} \cdot x''_{i,j}.
\label{ieq-cfl-overall-assignment-cost}
\end{align}
%where for any $i \in U$ and $j \in \MC{D}$, we define the absolute distance of any $p \in \Phi'(i,j) \cup \Phi''(i,j)$ as
%\mong{*** $\Phi''(i,j)$ only? ***}
%$$|p| \; := \; \sum_{\substack{i' \in \MC{F}, \; j' \in \MC{D}, \\[2pt] (j'^s,i') \in p}} c_{i',j'} \; + \; \sum_{\substack{ i' \in \MC{F}, \; j' \in \MC{D}, \\[2pt] (i',j'^s) \in p}} c_{i',j'}. $$
%
%
%Intuitively, i
In~(\ref{ieq-cfl-overall-assignment-cost}), we decompose the assignment cost of $\bm{x}'''$ into two parts, namely, (i) the assignment cost made by $\bm{f}$, $\bm{g}$, and $\bm{h}$ to facilities in $U$,
% with the cost incurred by conjugating pairs deducted 
and (ii) the assignment cost incurred by our iterative rounding process 
%rerouting the flow to sink via
to facilities in $I \setminus F'$.
%
In the following we consider the two parts separately.
%

\smallskip

%
We begin with the cost incurred by assignments to facilities in $I \setminus F'$. 
%
For any $i \in I \setminus F'$, let $(\bm{\lambda}^{\dagger(i)},\bm{\beta}^{\dagger(i)},\bm{\Gamma}^{\dagger(i)},\bm{\eta}^{\dagger(i)})$ be an optimal dual solution for~\ref{LP-natural-modified-dual} on $\Psi^{(i)}$.
%
The following lemma, which is proven by standard complementary slackness conditions, establishes the equivalence between the cost incurred around a non-extremal facility and the dual values it receives.
%\mong{*****}
%


\begin{lemma} \label{lemma-cfl-approx-primal-dual-for-I}
%
For any $i \in I \setminus F'$ and any $k \in F'^{(i)}$ with $0 < y^{\dagger(i)}_k < (1-\alpha)/2$, we have
$$\sum_{j \in D'^{(i)}} \lambda^{\dagger(i)}_j \cdot x^{\dagger(i)}_{k,j} \; = \; o_k \cdot y^{\dagger(i)}_k \; + \; \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j}. $$
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cfl-approx-primal-dual-for-I}]
%
Consider the complementary slackness conditions which follow from the fact that $(\bm{x}^{\dagger(i)},\bm{y}^{\dagger(i)})$ and $(\bm{\lambda}^{\dagger(i)},\bm{\beta}^{\dagger(i)},\bm{\Gamma}^{\dagger(i)},\bm{\eta}^{\dagger(i)})$ are optimal primal and dual solutions for~\ref{LP-natural-CFL-modified} and~\ref{LP-natural-modified-dual}, respectively.
%, we have \mong{*** polish ***}
Since $x^{\dagger(i)}_{k,j} > 0$ implies that corresponding dual inequality~\ref{cons-DM-1}, $\lambda_j \le \beta_k + \Gamma_{k,j} + c_{k,j}$, must be tight,
we have
$$\sum_{j \in D'^{(i)}} \; \lambda^{\dagger(i)}_j \cdot x^{\dagger(i)}_{k,j} \;\; = \;\; \beta^{\dagger(i)}_k \cdot \sum_{j\in D'^{(i)}} x^{\dagger(i)}_{k,j} \; + \; \sum_{j \in D'^{(i)}} \Gamma^{\dagger(i)}_{k,j} \cdot x^{\dagger(i)}_{k,j} \; + \; \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j}.$$
%
$\beta^{\dagger(i)}_k > 0$ and $\Gamma^{\dagger(i)}_{k,j} > 0$ imply that the corresponding inequalities~\ref{cons-PM-2} and~\ref{cons-PM-3} hold with equality.
%
Hence the above equality becomes
$$\beta^{\dagger(i)}_k \cdot u_k \cdot y^{\dagger(i)}_k \; + \; \frac{2\alpha}{1-\alpha} \sum_{j \in D'^{(i)}} r_j \cdot y^{\dagger(i)}_k \cdot\Gamma^{\dagger(i)}_{k,j} \; + \; \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j}.$$
%
The assumption that $y^{\dagger(i)}_k > 0$ and $y^{\dagger(i)}_k < (1-\alpha)/2$ imply that the corresponding inequality~\ref{cons-DM-2} holds with equality and the dual variable $\eta^{\dagger(i)}_k$ must be zero.
%
The above equality becomes
$$y^{\dagger(i)}_k \cdot \left( \; u_k \cdot \beta^{\dagger(i)}_k \; + \; \frac{2\alpha}{1-\alpha} \sum_{j \in D'^{(i)}} r_j \cdot \Gamma^{\dagger(i)}_{k,j} \; \right) \; + \; \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j} \; = \; o_k \cdot y^{\dagger(i)}_k \; + \; \sum_{j\in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j}, $$
and this lemma is proved.
%
%\begin{align*}
%& \sum_{j \in D'^{(i)}} \; \lambda^{\dagger(i)}_j \cdot x^{\dagger(i)}_{k,j} \\[4pt]
%= \;\; & \beta^{\dagger(i)}_k \cdot \sum_{j\in D'^{(i)}} x^{\dagger(i)}_{k,j} \; + \; \sum_{j \in D'^{(i)}} \Gamma^{\dagger(i)}_{k,j} \cdot x^{\dagger(i)}_{k,j} \; + \; \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j} & & 
%x^{\dagger(i)}_{k,j} > 0 \Rightarrow \text{\ref{cons-DM-1} tight} \notag 
%\\[6pt]
%= \;\; & y^{\dagger(i)}_k \cdot \left( \; u_k \cdot \beta^{\dagger(i)}_k \; + \; \frac{2\alpha}{1-\alpha} \sum_{j \in D'^{(i)}} r_j \cdot \Gamma^{\dagger(i)}_{k,j} \; \right) \; + \; \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j} & & 
%	\begin{matrix*}[l]
%	\beta^{\dagger(i)}_k > 0 \Rightarrow \text{\ref{cons-PM-2} tight} \\[3pt]
%	\Gamma^{\dagger(i)}_{k,j} > 0 \Rightarrow \text{\ref{cons-PM-3} tight}
%	\end{matrix*} \notag 
%\\[6pt]
%= \;\; & o_k \cdot y^{\dagger(i)}_k \; + \; \sum_{j\in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j}, & &
%	\begin{matrix*}[l]
%	y^{\dagger(i)}_k > 0 \Rightarrow \text{\ref{cons-DM-2} tight}, \\[3pt]
%	y^{\dagger(i)}_k < 1 \Rightarrow \eta^{\dagger(i)}_k = 0.
%	\end{matrix*}
%\end{align*}
%
%where in the first equality we use the fact that $x^{\dagger(i)}_{k,j} > 0$ implies that corresponding inequality $\lambda_j \le \beta_k + \Gamma_{k,j} + c_{k,j}$ is tight, in the second equality we use the fact that $\beta^{\dagger(i)}_k > 0$ and $\Gamma^{\dagger(i)}_{k,j} > 0$ imply that the corresponding inequalities~\ref{cons-PM-2} and~\ref{cons-PM-3} hold with equality, and in the last equality we use the fact that $y^{\dagger(i)}_k > 0$ and $y^{\dagger(i)}_k < (1-\alpha)/2$ imply that the corresponding inequality~\ref{cons-DM-2} holds with equality and the corresponding dual variable $\eta^{\dagger(i)}_k$ must be zero.
%
\end{proof}

%

%
The following lemma bounds the cost incurred by each individual facility $i \in I \setminus F'$ selected in our iterative rounding process.
%and the extra cost incurred for rerouting flows to sink via $i$.
%assignment cost incurred by rerouting the flow to sink via facilities in $I \setminus F'$ in our rounding process.
%

%

\begin{lemma} \label{lemma-cfl-cost-aggregation-per-iteration}
%
For any $i \in I \setminus F'$, we have 
%
\begin{align*}
(1-\alpha) \cdot o_i \; + \; \sum_{j \in D'^{(i)}} c_{i,j} \cdot x''_{i,j} \;\; \le \;\; 3\cdot \sum_{k \in F'^{(i)}} \; \sigma^{(i)}_k \cdot \left( \; o_k \cdot y^{\dagger(i)}_k \; + \; \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j} \; \right),
\end{align*}
where 
%in the above inequality 
we extend the definition of $\bm{\sigma}^{(i)}$ and define $\sigma^{(i)}_i := 1$ for notational brevity.
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cfl-cost-aggregation-per-iteration}]
%
If $y^{\dagger(i)}_i = (1-\alpha)/2$ holds in the beginning of the iteration when $i$ is selected for rounding, then $\sigma^{(i)}_k = 0$ for all $k \in F'^{(i)} \setminus \{i\}$ and this lemma holds trivially.
%

\smallskip

%
In the following we assume that $y^{\dagger(i)}_i < (1-\alpha)/2$.
%
Note that this implies that $y^{\dagger(i)}_k < (1-\alpha)/2$ for all $k \in F'^{(i)}$ by the design of the algorithm.
%
By the algorithm design, we have $\sum_{j \in D'^{(i)}} x^{\dagger(i)}_{i,j} > 0$, which implies that $y^{\dagger(i)}_i > 0$.
%
%
%\smallskip
%
%
Apply the definition of $\bm{x}''$ and the triangle inequality, we have
\begin{align}
\sum_{j \in D'^{(i)}} c_{i,j} \cdot x''_{i,j} \; 
= & \; \sum_{j \in D'^{(i)}} c_{i,j} \cdot \sum_{k \in F'^{(i)}} \sigma^{(i)}_k \cdot x^{\dagger(i)}_{k,j}  \;\;
%\notag \\[8pt]
\le \; \sum_{j \in D'^{(i)}, \; k \in F'^{(i)}} \sigma^{(i)}_k \cdot \left( c_{i,k} + c_{k,j} \right) \cdot x^{\dagger(i)}_{k,j}  \notag \\[8pt]
= & \; \sum_{j \in D'^{(i)}, \; k \in F'^{(i)}} \sigma^{(i)}_k \cdot c_{i,k} \cdot x^{\dagger(i)}_{k,j} \; + \; \sum_{k \in F'^{(i)}, \; j \in D'^{(i)}} \sigma^{(i)}_k \cdot c_{k,j} \cdot x^{\dagger(i)}_{k,j} .
\label{ieq-cfl-cost-aggregation-per-iteration-1}
\end{align}
%
By~(\ref{ieq-cfl-cost-aggregation-per-iteration-1}), to prove this lemma, it suffices to show that
\begin{align}
& (1-\alpha) \cdot o_i \; + \hspace{-4pt} \sum_{j \in D'^{(i)}, k \in F'^{(i)}} \hspace{-8pt} \sigma^{(i)}_k \cdot c_{i,k} \cdot x^{\dagger(i)}_{k,j} \;  
%\notag \\[2pt]
%& \hspace{2.8cm} 
\le \sum_{k \in F'^{(i)}} \sigma^{(i)}_k \cdot \left(  3\cdot o_k \cdot y^{\dagger(i)}_k  +  2 \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j} \right).
\label{ieq-cfl-cost-aggregation-per-iteration-2}
\end{align}
%

\smallskip

%
In the following we prove Inequality~(\ref{ieq-cfl-cost-aggregation-per-iteration-2}).
%
First, since $\left(\bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right)$ and $(\bm{\lambda}^{\dagger(i)}, \bm{\beta}^{\dagger(i)}, \bm{\Gamma}^{\dagger(i)}, \bm{\eta}^{\dagger(i)})$ are optimal primal and dual solutions for~\ref{LP-natural-CFL-modified} and~\ref{LP-natural-modified-dual}, it follows that, 
\begin{align}
\lambda^{(i)}_j = \beta^{(i)}_k + \Gamma^{(i)}_{k,j} + c_{k,j} \ge c_{k,j} \quad \text{for all $k \in F'^{(i)}, \; j \in D'^{(i)}$ with $x^{\dagger(i)}_{k,j} > 0$}.
\label{ieq-cfl-cost-aggregation-per-iteration-pd}
\end{align}
%$$\text{for any $k \in F'^{(i)}, \; j \in D'^{(i)}$,} \quad x^{\dagger(i)}_{k,j} > 0 \quad \text{implies that} \quad \lambda^{(i)}_j = \beta^{(i)}_k + \Gamma^{(i)}_{k,j} + c_{k,j} \ge c_{k,j}.$$
%
Therefore, applying the definition of $\sigma^{(i)}_k$, the fact that $c_{i,i} = 0$, and the triangle inequality,
we have 
\begin{align}
& \sum_{j \in D'^{(i)}, \; k \in F'^{(i)}} \sigma^{(i)}_k \cdot c_{i,k} \cdot x^{\dagger(i)}_{k,j} \;\; = \;\; \sum_{j \in D'^{(i)}, \; k \in F'^{(i)} \setminus \{i\}} \; \frac{1}{\sum_{\ell \in D'^{(i)}} x^{\dagger(i)}_{k,\ell}} \cdot \sum_{\ell \in D'^{(i)}} \sigma^{(i)}_{k,\ell} \cdot c_{i,k} \cdot x^{\dagger(i)}_{k,j}  \notag \\[8pt]
& \hspace{1.2cm} = \;\; \sum_{k \in F'^{(i)} \setminus \{i\}, \; \ell \in D'^{(i)}} c_{i,k} \cdot \sigma^{(i)}_{k,\ell} \;\; \le \;\; \sum_{k \in F'^{(i)} \setminus \{i\}, \; \ell \in D'^{(i)}} \left( \; c_{i,\ell} + c_{k,\ell} \; \right) \cdot \sigma^{(i)}_{k,\ell}  \;\;  \notag \\[8pt]
& \hspace{1.2cm} \le \;\; \sum_{k \in F'^{(i)} \setminus \{i\}, \; \ell \in D'^{(i)}} \left( \; c_{i,\ell} + \lambda^{(i)}_\ell \; \right) \cdot \sigma^{(i)}_{k,\ell},
\label{ieq-cfl-cost-aggregation-per-iteration-2-1}
\end{align}
where in the last inequality we apply the bound obtained in~(\ref{ieq-cfl-cost-aggregation-per-iteration-pd}).
%

\smallskip

%
Further applying the definition of $\sigma^{(i)}_{k,\ell}$ for all $k \in F'^{(i)} \setminus \{i\}$ and $\ell \in D'^{(i)}$, we obtain
\begin{align}
& \sum_{k \in F'^{(i)} \setminus \{i\}, \; \ell \in D'^{(i)}} \left( \; c_{i,\ell} + \lambda^{(i)}_\ell \; \right) \cdot \sigma^{(i)}_{k,\ell} \;\; =  \sum_{k \in F'^{(i)} \setminus \{i\}, \; \ell \in D'^{(i)}} \left( \; c_{i,\ell} + \lambda^{(i)}_\ell \; \right) \cdot \frac{x^{\dagger(i)}_{k,\ell}}{\sum_{p \in F'^{(i)} \setminus\{i\}} x^{\dagger(i)}_{p,\ell}} \cdot \delta^{(i)}_\ell  \notag \\[10pt]
& \hspace{1.2cm} = \;\; \sum_{\ell \in D'^{(i)}} \left( \; c_{i,\ell} + \lambda^{(i)}_\ell \; \right) \cdot \delta^{(i)}_\ell  \;\; \le \;\; \frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} \cdot \sum_{j \in D'^{(i)}} \left( \; c_{i,j} + \lambda^{(i)}_j \; \right) \cdot x^{\dagger(i)}_{i,j},
\label{ieq-cfl-cost-aggregation-per-iteration-3}
\end{align}
where in the last inequality we apply the definition of $\delta^{(i)}_\ell$ for all $\ell \in D'^{(i)}$.
%
%
Combining~(\ref{ieq-cfl-cost-aggregation-per-iteration-2-1}) and~(\ref{ieq-cfl-cost-aggregation-per-iteration-3}), we obtain
\begin{align}
& (1-\alpha) \cdot o_i \; + \hspace{-10pt} \sum_{j \in D'^{(i)}, k \in F'^{(i)}} \hspace{-8pt} \sigma^{(i)}_k \cdot c_{i,k} \cdot x^{\dagger(i)}_{k,j} \;  
%\notag \\[4pt]
%& \hspace{1.2cm} 
\le \; \frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} \cdot \left( 2\cdot o_i \cdot y^{\dagger(i)}_i + \sum_{j \in D'^{(i)}} \left( c_{i,j} + \lambda^{(i)}_j \right) \cdot x^{\dagger(i)}_{i,j} \right)  \notag \\[4pt]
& \hspace{4.2cm} = \;\; \frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} \cdot \left( \; 3 \cdot o_i \cdot y^{\dagger(i)}_i \; + 2\cdot \sum_{j \in D'^{(i)}} c_{i,j} \cdot x^{\dagger(i)}_{i,j} \; \right)  \notag \\[6pt]
& \hspace{4.2cm} = \;\; \frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} \cdot \theta \left( i, D'^{(i)}, \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right) \cdot \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{i,j},
\label{ieq-cfl-cost-aggregation-per-iteration-4}
\end{align}
where in the second last equality we apply Lemma~\ref{lemma-cfl-approx-primal-dual-for-I} with the fact that $0 < y^{\dagger(i)}_i < (1-\alpha)/2$, and in the last equality we apply 
%follows from Lemma~\ref{lemma-cfl-approx-primal-dual-for-I} and 
the definition of $\theta \left( i, D'^{(i)}, \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right)$.
%

\smallskip

%
By Lemma~\ref{lemma-approx-cfl-I-sparsely-loaded} and the definition of $\bm{x}''$, we have
\begin{align}
\frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} \cdot \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{i,j} \;\; = \;\; \sum_{j \in D'^{(i)}} x''_{i,j} \;\; = \;\; \sum_{j \in D'^{(i)}, \; k \in F'^{(i)}} \sigma^{(i)}_k \cdot x^{\dagger(i)}_{k,j}.
\label{ieq-cfl-cost-aggregation-per-iteration-5}
\end{align}
%
Moreover, by the design of the rounding algorithm, for any $k \in F'^{(i)}$, we have $$\theta \left( i, D'^{(i)}, \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right) \le \theta \left( k, D'^{(i)}, \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right).$$
%
Combining this property with~(\ref{ieq-cfl-cost-aggregation-per-iteration-5}), we have
\begin{align}
& \frac{1-\alpha}{2} \cdot \frac{1}{y^{\dagger(i)}_i} \cdot \theta \left( i, D'^{(i)}, \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right) \cdot \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{i,j}  \notag \\[6pt]
%& \hspace{0.6cm} = \;\; \sum_{k \in F'^{(i)}} \sigma^{(i)}_k \cdot \theta \left( i, D'^{(i)}, \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right) \cdot \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{k,j}  \notag \\[6pt]
& \hspace{0.6cm} \le \;\; \sum_{k \in F'^{(i)}} \sigma^{(i)}_k \cdot \theta \left( k, D'^{(i)}, \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right) \cdot \sum_{j \in D'^{(i)}} x^{\dagger(i)}_{k,j}  \notag \\
%& \hspace{0.6cm} = \;\; \sum_{k \in F'^{(i)}} \; \sigma^{(i)}_k \cdot \left( \;  o_k \cdot y^{\dagger(i)}_k \; + \; 2\cdot \sum_{j \in D'^{(i)}} \lambda^{\dagger(i)}_j \cdot x^{\dagger(i)}_{k,j} \; \right), \notag  \\
& \hspace{0.6cm} = \;\; \sum_{k \in F'^{(i)}} \; \sigma^{(i)}_k \cdot \left( \;  3\cdot o_k \cdot y^{\dagger(i)}_k \; + \; 2\cdot \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j} \; \right), 
\label{ieq-cfl-cost-aggregation-per-iteration-6}
\end{align}
where in the last inequality we apply the definition of $\theta \left( k, D'^{(i)}, \bm{x}^{\dagger(i)}, \bm{y}^{\dagger(i)} \right)$ 
%and the conclusion of Lemma~\ref{lemma-cfl-approx-primal-dual-for-I} 
for all $k \in F'^{(i)}$ with $\sum_{j \in D'^{(i)}} x^{\dagger(i)}_{k,j} > 0$.
%
%\smallskip
%
Combining~(\ref{ieq-cfl-cost-aggregation-per-iteration-4}) and~(\ref{ieq-cfl-cost-aggregation-per-iteration-6}) proves Inequality~(\ref{ieq-cfl-cost-aggregation-per-iteration-2}) and proves this lemma.
%
\end{proof}

%

\smallskip

%
Applying Lemma~\ref{lemma-cfl-approx-iterative-rounding-initial-tuple-feasibility}, Lemma~\ref{lemma-cfl-approx-iterative-rounding-successive-tuple-feasibility}, and Lemma~\ref{lemma-cfl-cost-aggregation-per-iteration}, we have the following lemma which establishes the guarantee for our iterative rounding process.
%

\begin{lemma} \label{lemma-cfl-cost-aggregation-overall}
%
\begin{align}
& \sum_{i \in I \setminus F'} \left( \; o_i \cdot y^*_i \; + \; \frac{1}{1-\alpha} \cdot \sum_{j \in D'^{(i)}} c_{i,j} \cdot x''_{i,j} \;\; \right)  \label{ieq-cfl-cost-aggregation-overall-target} \\[12pt]
& \hspace{3.2cm} \le \;\; \frac{3}{2 \cdot \alpha} \cdot \sum_{i \in I} \; o_i \cdot y'_i \; + \; \frac{3}{1-\alpha} \cdot \sum_{i \in I, \; j \in \MC{D}, \; p \in P(i,j)} |p| \cdot f_p.  \notag
\end{align}
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cfl-cost-aggregation-overall}]
%
%Consider the initial parameter tuple $\Psi^{(0)} = \left( F'^{(0)}, D'^{(0)}, r'^{(0)} \right)$ the algorithm maintains in the beginning of the iterative rounding process and l
Let $(\bm{x}^{*\dagger(0)}, \bm{y}^{*\dagger(0)})$ be an optimal solution for LP~\ref{LP-natural-CFL-modified} on the initial parameter tuple $\Psi^{(0)}$.
%
By Lemma~\ref{lemma-cfl-approx-iterative-rounding-initial-tuple-feasibility} and triangle inequality, it follows that 
\begin{align}
\sum_{i \in I} o_i \cdot y^{*\dagger(0)}_i \; + \; \sum_{i \in I, \; j \in \MC{D}} c_{i,j} \cdot x^{*\dagger(0)}_{i,j} \;\;
\le & \;\; \sum_{i \in I} o_i \cdot y^{\dagger(0)}_i \; + \; \sum_{i \in I, \; j \in \MC{D}} c_{i,j} \cdot x^{\dagger(0)}_{i,j}  \notag \\[6pt]
\le & \;\; \frac{1-\alpha}{2\cdot \alpha} \cdot \sum_{i \in I} o_i \cdot y'_i \; + \; \sum_{i \in I, \; j \in \MC{D}} \; \sum_{p \in P(i,j)} |p| \cdot f_p.
\label{ieq-cfl-cost-aggregation-overall-1}
\end{align}
%\mong{*** remove? ***} where for any $i \in \MC{F}, j \in \MC{D}$ and $p \in P(i,j)$, $|p|$ is absolute distance of path $|p|$ defined as
%$$|p| \; := \; \sum_{\substack{i' \in \MC{F}, \; j' \in \MC{D}, \\[2pt] (j'^s,i') \in p}} c_{i',j'} \; + \; \sum_{\substack{ i' \in \MC{F}, \; j' \in \MC{D}, \\[2pt] (i',j'^s) \in p}} c_{i',j'}. $$
%

\smallskip

%
For any $i \in I \setminus F'$, 
%consider 
%any iteration of the rounding process, say, 
%the iteration for which $i$ is selected for rounding.
%
%Let
let $\Psi''^{(i)} = (F''^{(i)}, D''^{(i)},\bm{r''}^{(i)})$ be the updated parameter tuple the algorithm maintains at the end of the iteration for rounding $i$,
%and the updated parameter tuple $\Psi''^{(i)} = (F''^{(i)}, D''^{(i)},\bm{r''}^{(i)})$ the algorithm maintains at the end of the iteration.
%
%Let
and $( {\bm{x}'}^{*\dagger(i)}, {\bm{y}'}^{*\dagger(i)} )$ be an optimal solution for~\ref{LP-natural-CFL-modified} on $\Psi''^{(i)}$.
%
%
%\smallskip
%
%
By Lemma~\ref{lemma-cfl-approx-iterative-rounding-successive-tuple-feasibility}, it follows that
\begin{align*}
& \sum_{k \in F''^{(i)}} o_k \cdot {y'}^{*\dagger(i)}_k + \hspace{-8pt} \sum_{k \in F''^{(i)}, \; j \in D''^{(i)}} c_{k,j} \cdot {x'}^{*\dagger(i)}_{k,j} \;
\le \; \sum_{k \in F''^{(i)}} \left( 1-\sigma^{(i)}_k \right) \cdot \left( o_k \cdot y^{\dagger(i)}_k + \sum_{j \in D''^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j} \right).
\end{align*}
%
%\smallskip
%
%
Combining the above with Lemma~\ref{lemma-cfl-cost-aggregation-per-iteration} and apply the fact that $F''^{(i)} = F'^{(i)} \setminus \{i\}$,
% and applying the definition of $\left( {\bm{x}'}^{\dagger(i)}, {\bm{y}'}^{\dagger(i)} \right)$, 
we have
\begin{align}
& \left( \; (1-\alpha) \cdot o_i \; + \; \sum_{j \in D'^{(i)}} c_{i,j} \cdot x''_{i,j} \; \right) \; + \; 3\cdot \left( \; \sum_{i \in F''^{(i)}} o_i \cdot {y'}^{*\dagger(i)}_i \; + \hspace{-8pt} \sum_{i \in F''^{(i)}, \; j \in D''^{(i)}} c_{i,j} \cdot {x'}^{*\dagger(i)}_{i,j} \; \right)  \notag \\[6pt]
& \hspace{1.2cm} \le \;\; 3 \cdot \sum_{k \in F'^{(i)}} \; \left( \; o_k \cdot y^{\dagger(i)}_k \; + \; \sum_{j \in D'^{(i)}} c_{k,j} \cdot x^{\dagger(i)}_{k,j} \; \right).
\label{ieq-cfl-cost-aggregation-overall-2}
\end{align}
%
Inequality~(\ref{ieq-cfl-cost-aggregation-overall-2}) shows that, the total cost incurred by $i$ can be bounded within three times the difference between the optimal values of the successive iterations.
%
%
%\smallskip
%
%
Taking the summation over $i \in I \setminus F'$ and applying 
%the fact that $(\bm{x}^{*\dagger(0)}, \bm{y}^{*\dagger(0)})$ is the optimal solution for the initial tuple $\Psi^{(0)}$, we obtain
Inequality~(\ref{ieq-cfl-cost-aggregation-overall-1}), we obtain
\begin{align*}
& \sum_{i \in I \setminus F'} \left( \; (1-\alpha) \cdot o_i \; + \; \sum_{j \in D'^{(i)}} c_{i,j} \cdot x''_{i,j} \;\; \right) \;\; \le \;\; 3\cdot \left( \; \sum_{i \in I} o_i \cdot y^{*\dagger(0)}_i \; + \; \sum_{i \in I, j \in \MC{D}} c_{i,j} \cdot x^{*\dagger(0)}_{i,j} \; \right) \\[12pt]
& \hspace{3.2cm} \le \;\; 3\cdot \frac{1-\alpha}{2 \cdot \alpha} \cdot \sum_{i \in I} \; o_i \cdot y'_i \; + \; 3 \cdot \sum_{i \in I, \; j \in \MC{D}, \; p \in P(i,j)} |p| \cdot f_p.
\end{align*}
%
%where the last inequality follows from .
%
Multiplying the above by $1/(1-\alpha)$ completes the proof of this lemma.
%
\end{proof}

%

%

\smallskip

%
Combining Lemma~\ref{lemma-cfl-cost-aggregation-overall} and Inequality~(\ref{ieq-cfl-overall-assignment-cost}), it follows that
\begin{align}
& \Psi\left( \; \frac{1}{1-\alpha} \cdot \bm{x}''', \; \bm{y}^* \; \right) \;\; = \;\; \sum_{i \in U \cup (I \setminus F')} \; o_i \cdot y^*_i \; + \; \frac{1}{1-\alpha} \cdot \sum_{i \in \MC{F}^*, \; j \in \MC{D}} c_{i,j} \cdot x'''_{i,j}  \notag \\[8pt]
& \hspace{1.6cm} \le \;\; \frac{1}{\alpha} \cdot \sum_{i \in U} \; o_i \cdot y'_i \; + \; \frac{1}{1-\alpha} \cdot \sum_{i \in U,\; j \in \MC{D}} \; \sum_{p \in \Phi'(i,j)} |p| \cdot f'_p  
\label{ieq-cfl-overall-bound-1} \\[6pt]
& \hspace{2.2cm} \; + \; \frac{3}{2\alpha} \cdot \sum_{i \in I} o_i \cdot y'_i \; \; + \; \frac{3}{1-\alpha} \cdot \sum_{i \in I, \; j \in \MC{D}, \; p \in P(i,j)} |p| \cdot f_p.
\label{ieq-cfl-overall-bound-2}
\end{align}
%
%
Recall 
%the fact 
that, in the definition of absolute length $|p|$ of any path $p \in \Phi'(i,j) \cup P(i,j)$, we take into account 
%only 
the length of the edges between $j'^s$ and $i'$ for any $j' \in \MC{D}$ and any $i' \in \MC{F}$.
%
Therefore, to bound the assignment cost incurred in~(\ref{ieq-cfl-overall-bound-1}) and~(\ref{ieq-cfl-overall-bound-2}), it suffices to consider the contribution of each possible pair of $j^s$ and $i$ for any $j \in \MC{D}$ and $i\in \MC{F}$.
% in~(\ref{ieq-cfl-overall-bound-1}).
%

\smallskip

%
We have the following lemma.
%

\begin{lemma}
\label{lemma-cfl-cost-overall-assignment-cost}
We have
\begin{align*}
& \frac{1}{1-\alpha} \cdot \sum_{\substack{ i \in U,\; j \in \MC{D}, \\[3pt] p \in \Phi'(i,j)}} |p| \cdot f'_p \;\; + \;\; \frac{3}{1-\alpha} \cdot \sum_{\substack{i \in I, \; j \in \MC{D}, \\[3pt] p \in P(i,j)}} |p| \cdot f_p  
\;\; \le \;\; \frac{7-4\alpha}{\left(1-\alpha\right)^2} \cdot \sum_{i \in \MC{F}, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j}.
\end{align*}
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cfl-cost-overall-assignment-cost}]
%
%We prove by considering the contribution of each in the l.h.s. of the inequality.
%
Let $i \in \MC{F}$ and $j \in \MC{D}$ be an arbitrary pair to be analyzed.
%
For notational brevity, define $\vphantom{\text{\LARGE T}} P' := \bigcup_{i^* \in U, \; j^* \in \MC{D}} \; \Phi'(i^*,j^*)$ and $P^{(I)} := \bigcup_{i^* \in I, \; j^* \in \MC{D}} \; P(i^*,j^*)$ to denote the set of paths considered in the statement of this lemma.
%
\phantom{\vphantom{\text{\Large T}}}Furthermore, let
\begin{align*}
& a^{(\Rightarrow)}_{i,j} \; := \; \frac{1}{1-\alpha} \cdot \sum_{\substack{p \in P', \\[2pt] (j^s,i) \in p}} f'_p \; + \; \frac{3}{1-\alpha} \cdot \sum_{\substack{p \in P^{(I)}, \\[2pt] (j^s,i) \in p}} f_p
\quad\enskip \text{and} \quad\enskip  \\[8pt]
& b^{(\Leftarrow)}_{i,j} \; := \; \frac{1}{1-\alpha} \cdot \sum_{\substack{p \in P', \\[2pt] (i,j^s) \in p}} f'_p \; + \; \frac{3}{1-\alpha} \cdot \sum_{\substack{p \in P^{(I)}, \\[2pt] (i,j^s) \in p}} f_p
\end{align*}
denote the total flow that is sent on the edge $(j^s,i)$ and $(i,j^s)$ by $\bm{f}'$ and $\bm{f}$, respectively.
%

\smallskip

%
In the following we will argue that 
\begin{equation}
a^{(\Rightarrow)}_{i,j} \; + \; b^{(\Leftarrow)}_{i,j} \;\; \le \;\; \frac{7-4\alpha}{\left(1-\alpha\right)^2} \enskip .
\label{ieq-cfl-cost-overall-assignment-cost-left-right-per-edge}
\end{equation}
%
Note that~(\ref{ieq-cfl-cost-overall-assignment-cost-left-right-per-edge}) implies the statement of this lemma directly.
%

\smallskip

%
Depending on the category to which $i$ belongs, consider the following cases.
%
We also refer the reader to Section~\ref{subsec-proof-approx-cfl-p1}, page~\pageref{subsec-proof-approx-cfl-p1}, for the definition of the function $\bm{f}'$.
%
%\mong{**** the simple illustrating figure ****}
%

\begin{itemize}
%
	\item
		$i \in U^{(>)}$ is tightly-occupied.
		
		In this case, by the definition of $\bm{f}'$, we have 
		\begin{align*}
		& \sum_{\substack{p \in P', \\[2pt] (j^s,i) \in p}} f'_p \;\; = \;\; (1-\alpha) \cdot g_{i,j} \; + \; \sum_{\substack{i^* \in U^{(>)}, \; j^* \in D^{(p)}, \\[2pt] p \in P(i^*,j^*), \\[2pt] (j^s,i) \in p }} \vspace{-2pt} f_p \;\; + \; \sum_{\substack{i^* \in U^{(<)}, \; j^* \in \MC{D}, \\[2pt] p \in P(i^*,j^*), \\[2pt] (j^s,i) \in p }} \vspace{-2pt} f_p, \quad\enskip \text{and}  \\[10pt]
		%
		& \sum_{\substack{p \in P', \\[2pt] (i,j^s) \in p}} f'_p \;\; = \;\; \sum_{\substack{i^* \in U^{(>)}, \; j^* \in D^{(p)}, \\[2pt] p \in P(i^*,j^*), \\[2pt] (i,j^s) \in p }} \vspace{-2pt} f_p \;\; + \; \sum_{\substack{i^* \in U^{(<)}, \; j^* \in \MC{D}, \\[2pt] p \in P(i^*,j^*), \\[2pt] (i,j^s) \in p }} \vspace{-2pt} f_p.
		\end{align*}
		%
		It follows that
		%
		\vspace{-18pt}
		\begin{align*}
		a^{(\Rightarrow)}_{i,j} \; + \; b^{(\Leftarrow)}_{i,j} \;\;
		& \le \;\; g_{i,j} \; + \; \frac{3}{1-\alpha} \cdot \left( \; \sum_{\substack{p \in P, \\[2pt] (j^s,i) \in p}} f_p \; + \; \sum_{\substack{p \in P, \\[2pt] (i,j^s) \in p }} f_p \; \right)  \\[10pt]
		& \le \;\; g_{i,j} \; + \; \frac{3}{1-\alpha} \cdot \left( \; x'_{i,j} \; + \; g_{i,j} \; \right) \;\;\; \le \;\;\; \frac{7-4\alpha}{\; \left(1-\alpha\right)^2 \;} \cdot x'_{i,j} \; ,
		\end{align*}
		%
		where the second last inequality follows from the feasibility of $\bm{f}$ for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ and the last inequality follows from the fact that $g_{i,j} = h_{i,j} \; \le \; x'_{i,j} / (1-\alpha)$.
		
	\item
		$i \in U^{(>)}$ is not tightly-occupied, $i \in U^{(<)}$, or $i \in I$.
		
		In these cases, we have $g_{i,j} = 0$, which implies that $b^{(\Leftarrow)}_{i,j} = 0$.
		%
		Furthermore, 
		\begin{align*}
		& \sum_{\substack{p \in P', \\[2pt] (j^s,i) \in p}} f'_p \;\; \le \;\; (1-\alpha) \cdot h_{i,j} \; + \; \sum_{\substack{j^* \in \MC{D}, \; p \in P(i,j^*), \\[2pt] (j^s,i) \in p }} \vspace{-2pt} f_p \;\; \le \;\; 2\cdot x'_{i,j}
		\end{align*}
		%
		by the feasibility of $\bm{f}$ for $\mathbf{MFN}_\Psi(\bm{x}',\bm{y}'',\bm{g})$ and the fact that $h_{i,j} \; \le \; x'_{i,j} / (1-\alpha)$.
		%
		
		Hence, we have
		\vspace{-18pt}
		\begin{align*}
		a^{(\Rightarrow)}_{i,j} \; + \; b^{(\Leftarrow)}_{i,j} \;\;
		& \le \;\; \frac{5}{1-\alpha} \cdot x'_{i,j} \;\; < \;\; \frac{7-4\alpha}{\; \left(1-\alpha\right)^2 \;} \cdot x'_{i,j} \; .
		\end{align*}
%
\end{itemize}
%
In either case,~(\ref{ieq-cfl-cost-overall-assignment-cost-left-right-per-edge}) holds. 
%
This proves the lemma.
%
\end{proof}

\smallskip

%
\noindent
Combining~(\ref{ieq-cfl-overall-bound-1}) and~(\ref{ieq-cfl-overall-bound-2}) with Lemma~\ref{lemma-cfl-cost-overall-assignment-cost}, we obtain
\begin{align*}
\Psi\left( \; \frac{1}{1-\alpha} \cdot \bm{x}''', \; \bm{y}^* \; \right) \; 
& \le \;\; \frac{3}{2\alpha} \cdot \sum_{i \in \MC{F}} o_i \cdot y'_i \; + \; \frac{7-4\alpha}{\left( 1-\alpha \right)^2} \cdot \sum_{i \in \MC{F}, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j} \\[4pt]
& \le \;\; \max\left\{ \; \frac{3}{2\alpha} \; , \; \frac{7-4\alpha}{(1-\alpha)^2} \; \right\} \cdot \Psi(\bm{x}', \bm{y}').
\end{align*}
%
This completes the proof of Lemma~\ref{theorem-cfl-sparse-flow-approx} and proves Theorem~\ref{thm-MFN-relaxed-separation-oracle}.
%and this lemma is proved.

%

\newpage


%%%%
%%%%

%


%


\begin{figure*}[h]
\centering
\fbox{
\begin{minipage}{.8\textwidth}
\begin{align}
& \text{min} & & \sum_{i \in \MC{F}} \; y_i \; + \; \sum_{i \in \MC{F}, \hspace{2pt} j \in \MC{D}} c_{i,j}\cdot x_{i,j} & &  \label{LP-natural-CFL} \tag*{LP-(N)} \\[3pt]
& \text{s.t.} & & \sum_{i\in \MC{F}} \; x_{i,j} \; \ge \; 1, & & \forall j\in \MC{D} \label{cons-P-1} \tag*{(N-1)} \\[3pt]
& & & \sum_{j \in \MC{D}} \; x_{i,j} \; \le \; u_i \cdot y_i, & & \forall i \in \MC{F} \label{cons-P-2} \tag*{(N-2)} \\[3pt]
& & & 0 \; \le \; x_{i,j} \; \le \; y_i, & & \forall i \in \MC{F}, j \in \MC{D} \quad \label{cons-P-3} \tag*{(N-3)} \\[5pt]
& & & 0 \; \le \; y_i \; \le \; 1, & & \forall i \in \MC{F}. \label{cons-P-4} \tag*{(N-4)}
\end{align}
\vspace{-6pt}
\end{minipage}\quad
}
%
\fbox{
\begin{minipage}{.8\textwidth}
\begin{align}
& \text{max} & & \sum_{j \in \MC{D}} \; \alpha_j \; - \; \sum_{i \in \MC{F}} \; \eta_i & &  \label{LP-natural-dual} \tag*{LP-(DN)} \\[6pt]
& \text{s.t.} & & \alpha_j \; \le \; \beta_i \; + \; \Gamma_{i,j} \; + \; c_{i,j}, \enskip & & \forall i \in \MC{F}, j\in \MC{D}, & & \label{cons-D-1} \tag*{(D-1)} \\[6pt]
& & & u_i \cdot \beta_i \; + \; \sum_{j \in \MC{D}} \Gamma_{i,j} \; \le \; 1 \; + \; \eta_i, & & \forall i \in \MC{F},  \label{cons-D-2} \tag*{(D-2)} \\[1pt]
& & & \alpha_j, \; \beta_i, \; \Gamma_{i,j}, \; \eta_i \; \ge \; 0, & & \forall i \in \MC{F}, j \in \MC{D}. \label{cons-D-3} \tag*{(D-3)}
\end{align}
\vspace{-6pt}
\end{minipage}\quad
}
\caption{(Restate for further reference) The natural LP formulations for CFL-CFC.}
\label{fig-natural-LP}
\end{figure*}


%

%

%%%
%%%

\section{Proof of Theorem~\ref{theorem-cfl-ufc-approx}.}
\label{sec-proof-approx-cfl-cfc}

%
The proof of this theorem is outlined as follows.
%
First we define 
%in the following 
the notations and notions that are used throughout this section.
%
In Section~\ref{subsec-proof-approx-cfl-cfc-feasibility}, page~\pageref{subsec-proof-approx-cfl-cfc-feasibility}, we show that our rounding algorithm is well-defined and terminates in polynomial time.
%
We define in the same section an intermediate assignment $\bm{x}^\circ$ and establish the feasibility of $(\bm{x}^\circ, y^*)$ for 
%the input instance 
$\Psi$.
%
This shows that the feasible region of the min-cost assignment problem between $\MC{D}$ and $\MC{F}^*$ is nonempty.
%
%
%\smallskip
%
%
We establish the approximation guarantee for $(\bm{x}^\circ, y^*)$ in Section~\ref{subsec-proof-approx-cfl-cfc-guarantee}, page~\pageref{subsec-proof-approx-cfl-cfc-guarantee}.
% for $(\bm{x}^\circ, y^*)$.
%
%To prove this theorem,
%Theorem~\ref{theorem-cfl-ufc-approx},
%In the following, 
%we show that an intermediate assignment $\bm{x}^\circ$ defined from the rounding process together with $y^*$ forms a feasible solution for $\Psi$ and meets the statement of Theorem~\ref{theorem-cfl-ufc-approx}.
% with $y^*$.
%
%Note that t
This completes the proof since $x^\dagger$ is the optimal solution for the min-cost assignment problem between $\MC{D}$ and $\MC{F}^*$.
%
%\smallskip

%\medskip

%\vspace{-6pt}
%
\paragraph{Notations.}

Consider the cluster-forming process. 
%
Let $\MC{C}_{D'}$ and $\MC{C}_{H'}$ denote the sets of clusters centered at the non-outlier clients and outlier clients, respectively.
%
For each $q \in \MC{C}_{D'}$, we use $j(q)$ to denote the center client of $q$ 
%Note that $j(q) \in \MC{D} \cup H$.
%
%For each $q \in \MC{C}_{D'}$, further let
and $i(q)$ denote the facility that is selected to be rounded up in the iteration when $q$ is formed.
%
Let $F^*_{D'} := \left\{ \; \vphantom{\text{\Large T}} i(q) \; \colon \; q \in \MC{C}_{D'} \; \right\}$ denote the set of facilities rounded up for the clusters in $\MC{C}_{D'}$.
%
Note that, by the setting of the algorithm, $F^*_{D'}$ and $G$ are mutually exclusive, and the set of satellite facilities $B(j)$ for each $j \in H$ jointly forms a partition of $G$.
%

\smallskip

%
To prevent notational ambiguity, 
%\mong{****** initial or current ***** interchangeably when there is no confusion in the context ******} 
we will use $(\bm{x}', \bm{y}')$ to denote the initial solution the algorithm has for $\Psi$.
%For the outlier clients $j \in H$, 
%
%assignment of $j$ to the facilities in $I$ when it is created.
%
For each $q \in \MC{C}_{D'}$, we 
%refer to 
use $D'^{(q)}$, $F'^{(q)}$, $H^{(q)}$, $H'^{(q)}$, $\bm{x}'^{(q)}$, and $\bm{y}'^{(q)}$
%, $F'$, $D'$, and $H'$ as 
to denote the set $D'$, the set $F'$, the sets $H$, the set $H'$, the assignment $\bm{x}'$, and the multiplicity $\bm{y}'$
%, and the sets $F'$, $D'$, $H'$ 
the algorithm maintains at the moment when the cluster $q$ is formed.
%
We use $B(q)$ to denote the set $N_{(F',x')}(j(q))$ of satellite facilities at that moment.
%
Similarly, we use $\bm{x}'^{(\text{II})}$ and $\bm{y}'^{(\text{II})}$ to denote the assignment $\bm{x}'$ and the multiplicity $\bm{y}'$ the algorithm maintains when it enters the second phase.
%

%\smallskip

%
% and $(\tilde{\bm{x}}',\tilde{\bm{y}}')$ to denote the 
%current 
%solution the algorithm maintains at any particular moment, referred to by the context, during the rounding process.
%
%Similarly, we use $\tilde{F}'$, $\tilde{D}'$, and $\tilde{H}$ to denote the sets $F'$, $D'$, and $H$ the algorithm maintains at any particular moment referred to by the context during the rounding process.
%
%\mong{**** double check the consistency ****}
%We will 
%


\smallskip

%
%Consider the set $H$ of outlier clients.
%
For each outlier client $j \in H$, we use $w(j)$ to denote the facility in $U$ at which $j$ is located.
%
We use $p(j)$ to denote the specific parent client in $J^{(\leftrightarrow)}$ from which $j$ is created.
% and \mong{**** remove $s(j)$ ? *****} $s(j)$ to denote the set of outlier clients in $H$ that are created from $p(j)$.
%
%
%\smallskip
%
%
On the contrary, for any $w \in U$, we use $H(w)$ to denote the set of outlier clients located at $w$.
%
For any $j \in J^{(\leftrightarrow)}$, we use $H(j)$ to denote the set of outlier clients that are created from $j$.
%
%

\smallskip

%
For notational brevity, for any assignment function $\bm{x}$ of interest, we will use $\bm{x}|_{A,B}$ to denote the assignments made in $\bm{x}$ between $A \subseteq \MC{F}$ and $B \subseteq \MC{D}$.
%
Similarly, for any multiplicity function $\bm{y}$ of interest, we will use $\bm{y}|_A$ to denote the multiplicity of facilities in $A \subseteq \MC{F}$ in $\bm{y}$.
% and any $A \subseteq \MC{D}$, let $\bm{x}|_A$ denote the assignments made in $\bm{x}$ for the clients in $A$, i.e., the assignment function when restricted only to clients in $A$.
The value of zero is assumed for pairs and facilities not in the defining domain of $\bm{x}$ and $\bm{y}$.
%
%Similarly, for any $B \subseteq \MC{F}$, let $\bm{x}|_B$ and $\bm{y}_B$ denote the assignments
%
%\smallskip
%
%\mong{*******}
%
%For any assignment function $\bm{x}$, any $A \subseteq \MC{F}$, and any $B \subseteq \MC{D}$, we use $\bm{x}|_{A,B}$ denote the assignments made in $\bm{x}$ between clients in $A$ and facilities in $B$, i.e., the assignment function when restricted to $A \times B$.
%
%Similarly, for any multiplicity function $\bm{y}$, we use $\bm{y}|_B$ denote the multiplicities of the facilities in $B$.
%
%\mong{*******}
%
%\mong{**** zero outside the defining domain ****}
%




%\medskip

%

%%%
%%%

%\paragraph{Feasibility.}
\subsection{The Feasibility}
\label{subsec-proof-approx-cfl-cfc-feasibility}

%
%In the following w
We first consider 
%in page~\pageref{par-proof-approx-cfl-cfc-feasibility-cd} the
our rounding process for clusters in $\MC{C}_{D'}$ in the first phase and
show that the rounding algorithm is well-defined and runs in polynomial time.
%
%We  and
%.
%
We consider 
in page~\pageref{par-proof-approx-cfl-cfc-feasibility-ch} the rounding process for clusters in $\MC{C}_{H'}$ in the second phase.
%
In page~\pageref{par-proof-approx-cfl-cfc-feasibility-xc} we define the assignment $\bm{x}^\circ$ and show that $(\bm{x}^\circ, \bm{y}^*)$ forms a feasible solution for $\Psi$.
%

%\smallskip

%

%%%%
%%%%

\paragraph{The rounding process for $\MC{C}_{D'}$.}
\label{par-proof-approx-cfl-cfc-feasibility-cd}

In the following we consider the first phase of our rounding process.
%
To make the presentation precise, we will further use $(\tilde{\bm{x}}',\tilde{\bm{y}}')$, $\tilde{F}'$, $\tilde{D}'$, and $\tilde{H}$ to denote the 
%current 
solution $(\bm{x}', \bm{y}')$, the sets $F'$, $D'$, and $H$ the algorithm maintains at any particular moment, referred to by the context, during the rounding process in the first phase.
%

\smallskip

%
In order to characterize the behavior of the rounding process, we consider two types of constraints for 
%the solution 
$(\tilde{\bm{x}}',\tilde{\bm{y}}')$ 
%the algorithm maintains 
throughout the process, namely, 
%which regards 
the capacity constraints for facilities in $\tilde{F}'$ and constraint~\ref{cons-P-3} from~\ref{LP-natural-CFL} for facilities in $\tilde{F}'$ and clients in $\tilde{D}'$, listed as follows.

%
\begin{center}
\begin{minipage}{.56\textwidth}
\fbox{
\begin{minipage}{\textwidth}
\vspace{6pt}
\begin{align}
& \sum_{j \in \MC{D} \cup \tilde{H}} \; \tilde{x}'_{i,j} \; \le \; u_i \cdot \tilde{y}'_i, & & \forall i \in \tilde{F}'. \label{cons-P-2-with-H} \tag*{(MN-2)} \\[4pt]
& \hspace{1.3cm} \tilde{x}'_{i,j} \; \le \; \tilde{y}'_i, & & \forall i \in \tilde{F}', j \in \tilde{D}'.  \label{cons-P-3} \tag*{(N-3)}
\end{align}
\vspace{-4pt}
\end{minipage}\qquad
}
\end{minipage}
\end{center}
%

It is clear that~\ref{cons-P-2-with-H} and~\ref{cons-P-3} hold in the beginning of the rounding process since initially $F' := I$, $D' := J^{(I)} \cup J^{(\leftrightarrow)}$, $H := \emptyset$, and $(\tilde{\bm{x}}', \tilde{\bm{y}}')$ is a feasible solution for~\ref{LP-natural-CFL}.
%

\smallskip

%
The following two lemmas establish that, throughout the first phase of the rounding process, the constraints~\ref{cons-P-2-with-H} and~\ref{cons-P-3} remain valid with respect to 
%the current solution 
$(\tilde{\bm{x}}',\tilde{\bm{y}}')$,
% and the current 
$\tilde{F}'$, $\tilde{D}'$, and $\tilde{H}$.
%

%

\begin{lemma} \label{lemma-feasibility-creating-outlier-client}
The process of creating outlier clients 
%from a client $j \in \MC{D}$ 
does not render~\ref{cons-P-2-with-H} invalid.
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-feasibility-creating-outlier-client}]
%
Consider the process of creating outlier clients from a client, say, $j \in J^{(\leftrightarrow)} \cap D'$.
%
By the algorithm design, this happens when $\sum_{i \in \tilde{F}'} \tilde{x}'_{i,j} < 1/2$.
%

\smallskip

%
For each $i \in N_{(\tilde{F}',\tilde{x}')}(j)$, consider the assignments the algorithm has made from $H(j)$ to $i$ after $H(j)$ is created.
%
By the algorithm setting and the definition of $d_\ell$ for any $\ell \in H(j)$, we have
%
\begin{align*}
\sum_{\ell \in H(j)} x'_{i,\ell} \; := \; \sum_{\ell \in H(j)} d_\ell \cdot \frac{ \tilde{x}'_{i,j} }{ \sum_{ k \in \tilde{F}'} \tilde{x}'_{k,j} } 
\;\; = \;\; \frac{ \tilde{x}'_{i,j} }{ \sum_{ k \in \tilde{F}'} \tilde{x}'_{k,j} } \cdot \sum_{\ell \in H(j)} r'_j \cdot \frac{ \tilde{x}'_{w(\ell),j} }{ \sum_{w \in U} \tilde{x}'_{w,j} }.
\end{align*}
%
By the definition of $r'_j$, we have $r'_j \le \sum_{k \in \tilde{F}'} \tilde{x}'_{k,j}$.
%
Hence, the above becomes
\begin{align}
\sum_{\ell \in H(j)} x'_{i,\ell} \;\; \le \;\;  \tilde{x}'_{i,j} \cdot \sum_{\ell \in H(j)} \frac{ \tilde{x}'_{w(\ell),j} }{ \sum_{w \in U} \tilde{x}'_{w,j} } \;\; = \;\; \tilde{x}'_{i,j},
\label{ieq-lemma-feasibility-creating-outlier-client-1}
\end{align}
where in the last equality we apply the fact that $\sum_{\ell \in H(j)} \tilde{x}'_{w(\ell),j} = \sum_{w \in U} \tilde{x}'_{w,j}$.
%
Since the algorithm resets $\vphantom{\text{\Large T}} \tilde{x}_{i,j}$ to be zero after $H(j)$ is created, it follows from~(\ref{ieq-lemma-feasibility-creating-outlier-client-1}) that the LHS of~\ref{cons-P-2-with-H} for $i$ does not increase after $H(j)$ is created and the new assignments to $i$ are made.
%
\end{proof}

%

%We begin with the following lemma, which shows that the scaled-down operation in the rounding process is well-defined and keeps the LP constraints~\ref{cons-P-2} and~\ref{cons-P-3} valid throughout the process.
%
%\mong{***** take outlier clients into consideration *****}
%

%\smallskip

%

\begin{lemma} \label{lemma-feasibility-scaled-down-well-defined}
%
We have $0 < \delta_{i(q)} \le 1$ for any $q \in \MC{C}_{D'}$.
%
Furthermore, the scaled-down operation the algorithm performs when rounding $i(q)$ does not render~\ref{cons-P-2-with-H} and~\ref{cons-P-3} invalid.
%of~\ref{LP-natural-CFL} are valid throughout the rounding process.
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-feasibility-scaled-down-well-defined}]
%
Consider any $q \in \MC{C}_{D'}$.
%
We will show that, provided that constraints~\ref{cons-P-2-with-H} and~\ref{cons-P-3} are valid in the beginning of the iteration for which $q$ is formed,
we have $0 < \delta_{i(q)} \le 1$ and the rounding process for $i(q)$ does not render~\ref{cons-P-2-with-H} and~\ref{cons-P-3} invalid.
%
Note that this proves the lemma together with Lemma~\ref{lemma-feasibility-creating-outlier-client} since~\ref{cons-P-2-with-H} and~\ref{cons-P-3} are valid in the beginning of the process.
%

\smallskip

%
Since $i(q) \in F'^{(q)}$, we know that $y'^{(q)}_{i(q)} < 1/2$.
%
Furthermore, since $j(q)$ is selected as the center client and since $j(q) \in D'^{(q)}$ by assumption, it follows that $\sum_{i \in F'^{(q)}} x'^{(q)}_{i,j(q)} \ge 1/2$.
%
This implies that
$$\sum_{i \in B(q) \setminus \{i(q)\}} y'^{(q)}_i \; \ge \; \sum_{i \in B(q) \setminus \{i(q)\}} x'^{(q)}_{i,j(q)} \; \ge \; \frac{1}{2} - x'^{(q)}_{i(q),j(q)} \; \ge \; \frac{1}{2} - y'^{(q)}_{i(q)} \; > \; 0,$$
where in the above inequalities we apply constraint~\ref{cons-P-3} for $i \in B(q)$ and $j(q)$.
%
This shows that $$\delta_{i(q)} \; := \; \left( \; \frac{1}{2} - y'^{(q)}_{i(q)} \; \right) \cdot \frac{1}{ \sum_{i \in B(q)\setminus \{i(q)\}} y'^{(q)}_i } \; > \; 0.$$
%
On the contrary, by~\ref{cons-P-3} we have $\sum_{i \in B(q)} y'^{(q)}_i \; \ge \; \sum_{i \in B(q)} x'^{(q)}_{i,j(q)} \; \ge \; 1/2$.
%
This implies that $1/2 - y'^{(q)}_{i(q)} \le \sum_{i \in B(q) \setminus \{i\}} y'^{(q)}_i$ and $\delta_{i(q)} \le 1$.
% $$\delta_{i(q)} \; := \; \left( \; \frac{1}{2} - y'^{(q)}_{i(q)} \; \right) \cdot \frac{1}{ \sum_{i \in B(q)\setminus \{i(q)\}} y'^{(q)}_i } \; \le \; 1.$$
%

\smallskip

%
To see that constraints~\ref{cons-P-2-with-H} and~\ref{cons-P-3} remain valid at the end of this iteration, observe that for each $i \in B(q) \setminus \{i(q)\}$ and any $j \in D'^{(q)}$, both $x'^{(q)}_{i,j}$ and $y'^{(q)}_i$ are scaled down simultaneously by the constant $\left( 1-\delta_{i(q)}\right)$.
%
%Hence the constraints remain valid.
%
This completes the proof of this lemma.
%
\end{proof}

%
By Lemma~\ref{lemma-feasibility-creating-outlier-client} and Lemma~\ref{lemma-feasibility-scaled-down-well-defined}, we have the following corollary.
%

\begin{corollary} \label{cor-feasibility-cons}
Throughout the first phase of the rounding process, the constraints~\ref{cons-P-2-with-H} and~\ref{cons-P-3} remain valid with respect to $(\tilde{\bm{x}}', \tilde{\bm{y}}')$, $\tilde{F}'$, $\tilde{D}'$, and $\tilde{H}$.
\end{corollary}

%\begin{proof}[Proof of Corollary~\ref{cor-feasibility-cons}]
%
%\mong{***}
%
%\end{proof}

%
Lemma~\ref{lemma-feasibility-scaled-down-well-defined} shows that the scaled-down operation in our rounding process is well-defined.
%
%In the following we argue that the rounding process terminates in polynomial time.
%
Since constraint~\ref{cons-P-3} holds throughout the process, it follows that, for any particular moment, 
\begin{align}
\sum_{i \in \tilde{F}'} \tilde{x}'_{i,j} \;\; \le \;\; \sum_{i \in \tilde{F}'} \tilde{y}'_i \quad \text{holds for any $j \in \tilde{D}'$.}
\label{ieq-feasibility-cd-1}
\end{align}
%
By the algorithm design, we know that at least one facility is removed from $F'$ after each iteration in the first phase. 
%
Therefore, the rounding process repeats for at most $|F'| := |I|$ iterations before $F'$ becomes empty, which in turn implies that $D' \cup H'$ is empty by~(\ref{ieq-feasibility-cd-1}).
%
This shows that the rounding algorithm terminates in polynomial time.
%

\smallskip

%
The following lemma, which shows that the rounded facility is sparsely loaded by the rerouted assignments, is straightforward to verify.
%

\begin{lemma} \label{lemma-feasibility-absorbing-process}
%
We have $\sum_{j \in \MC{D} \cup H} x^*_{i(q),j} \le u_{i(q)} / 2$ for any $q \in \MC{C}_{D'}$.
%$i \in F^*_{D'}$,
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-feasibility-absorbing-process}]
%
Consider any $q \in \MC{C}_{D'}$.
%specific iteration for which a cluster $q \in \MC{C}_{D'}$ is formed and about to be rounded.
%
By the design of the algorithm and the fact that constraint~\ref{cons-P-3}
% of~\ref{LP-natural-CFL} 
holds throughout the process, we have
\begin{align*}
\sum_{j \in \MC{D} \cup H} x^*_{i(q),j} \;\; 
& = \;\; \sum_{j \in \MC{D} \cup H} x'^{(q)}_{i(q),j} \; + \; \sum_{i \in B(q) \setminus \{i(q)\}} \; \sum_{j \in \MC{D} \cup H} \delta_{i(q)} \cdot x'^{(q)}_{i,j} \\[4pt]
& \le \;\; u_{i(q)} \cdot y'^{(q)}_{i(q)} \; + \; \sum_{i \in B(q) \setminus \{i(q)\}} \delta_{i(q)} \cdot u_i \cdot y'^{(q)}_i \\[2pt]
& \le \;\; u_{i(q)} \cdot y'^{(q)}_{i(q)} \; + \; u_{i(q)} \cdot \left( \; \frac{1}{2} - y'^{(q)}_{i(q)} \; \right) \;\; = \;\; \frac{1}{2} \cdot u_{i(q)},
\end{align*}
where in the second last inequality we use apply fact that $u_{i(q)} \ge u_i$ for all $i \in B(q)$ by the way $i(q)$ is selected and the definition of $\delta_{i(q)}$.
%
\end{proof}

%

%

%%%%
%%%%

\paragraph{The rounding process for $\MC{C}_{H'}$.}
\label{par-proof-approx-cfl-cfc-feasibility-ch}

In the following we consider the rounding process for clusters in $\MC{C}_{H'}$ in the second phase.
%
%We begin with t
The following lemma summarizes the status of the 
%corresponding 
facilities and clients
%at the moment 
when the algorithm enters the this phase.
%The following two lemmas, which regard the behavior and properties of our rounding process for clusters in $\MC{C}_{D'}$, are straightforward to verify.
% shows that feasibility of \mong{***} for the satellite facilities is preserved in the rounding process for $\MC{C}_{D'}$. \mong{**** $x'^{(q)}, y'^{(q)}$ and $x'^{(\text{II})}, y'^{(\text{II})}$}



\begin{lemma} \label{lemma-assignment-absorbing-process}
%
When the algorithm enters the second phase, the following holds.
% for any $j \in \MC{D} \cup H$,
\begin{itemize}
	\item
		For any $i \in G$, \hspace{1pt} $\sum_{j \in \MC{D} \cup H} x'^{(\text{II})}_{i,j} \le u_i \cdot y'^{(\text{II})}_i$.

	\item
		For any $j \in J^{(I)}$, \hspace{1pt} $\sum_{i \in I} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} \; > \; 1/2$.
		
	\item
		For any $j \in J^{(\leftrightarrow)}$, \hspace{1pt} $\sum_{i \in I} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} + \sum_{i \in U} x'_{i,j} \; > \; 1/2$.
		
	\item
		For any $j \in H$, \hspace{1pt} $\sum_{i \in F^*_{D'}} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} \; = \; \sum_{i \in I} x'_{i,j}$.
\end{itemize}
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-assignment-absorbing-process}]
%
The first statement of this lemma follows directly from Corollary~\ref{cor-feasibility-cons} and the definition of $(\bm{x}'^{(\text{II})}, \bm{y}'^{(\text{II})})$.
% and the fact that constraint~\ref{cons-P-2} for each $i \in G$ is valid in the beginning of the algorithm.
%
%
%\smallskip
%
%
The remaining 
%statements 
of this lemma follows from the way how the algorithm handles the residue demand of each client.
%
Consider the moment for which each $j \in J^{(I)} \cup J^{(\leftrightarrow)} \cup H$ is removed from consideration in the first phase, and the fact that $G := \bigcup_{j \in H} B(j)$.
%$D' \cup H'$.
%

\smallskip

%
For $j \in J^{(I)} \cup J^{(\leftrightarrow)}$, it is removed when $\sum_{i \in F'} x'_{i,j} < 1/2$. 
%
It follows that the assignments rerouted to clusters in $\MC{C}_{D'}$, the assignments taken into clusters in $\MC{C}_{H'}$, and possibly the original assignments to facilities in $U$, account for at least $1/2$.
%, i.e., $\sum_{i \in I} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} > 1/2$. 
%
%Similar argument holds for $j \in J^{(\leftrightarrow)}$.
%

\smallskip

%
For $j \in H$, it is removed when selected as the center of a cluster that is possibly empty.
%
When this happens, all of the remaining assignments for $j$ are taken into this cluster.
%
\end{proof}

%

\smallskip

%
Lemma~\ref{lemma-assignment-absorbing-process} leads to the following corollary.
% follows directly from Lemma~\ref{lemma-assignment-absorbing-process}.

\begin{corollary} \label{cor-scaling-factor}
%
$0 \le t'_j \le 2$ for all $j \in \MC{D}$.
%
\end{corollary}

\begin{proof}[Proof of Corollary~\ref{cor-scaling-factor}]
%
%First we prove that $t'_j \le 2$ for all $j \in \MC{D}$.
%
It suffices to prove the statement for $j \in \MC{D}$ with $\sum_{i \in I} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} > 0$.
%
Since $\sum_{i \in I}x^*_{i,j}  + \sum_{i \in G} x'^{(\text{II})}_{i,j} > 0$ implies that $j \in J^{(I)} \cup J^{(\leftrightarrow)}$, by the definition of $r'_j$, we have
$$1-\sum_{i \in U} x'_{i,j} - r'_j \;\; \ge \;\; 1-\sum_{i \in U} x'_{i,j} - \sum_{i \in I} x'_{i,j} \;\; \ge \;\; 0, \enskip \text{which implies that} \enskip t'_j > 0.$$
%
%Moreover, $\sum_{i \in I} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} \ge \sum_{i \in I} x^*_{i,j} > 0$.
%
%It follows that $t'_j \ge 0$.
%

\smallskip

%
In the following we show that $t'_j \le 2$.
%Consider the definition of $t'_j$ for any $j \in \MC{D}$.
%
Since $j \in J^{(I)} \cup J^{(\leftrightarrow)}$, it suffices to prove the statement for the following cases.
%
\begin{itemize}
	\item
		If $j \in J^{(I)}$, then $t'_j < 2$ directly from the conclusion of Lemma~\ref{lemma-assignment-absorbing-process} since $1\sum_{i \in I} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} > 1/2$ by the algorithm design and $(1-\sum_{i \in U}x'_{i,j} - r'_j) \le 1$.
		
		%then $\sum_{i \in I}x^*_{i,j} > 0$ implies that $\sum_{i \in I} x^*_{i,j} > 1/2$ when the algorithm enters the second phase. 
		%
		%Hence $\vphantom{\text{\LARGE T}} t'_j \; \le \; 1 / \sum_{ i \in I} x^*_{i,j} \; < \; 2$.
		
	\item
		If $j \in J^{(\leftrightarrow)}$ and $r'_j \neq \sum_{i \in U} x'_{i,j}$,
		%
		then all of the residue demand of $j$ has been redistributed as outlier clients when $j$ is to be removed from $D'$.
		%
		It follows that $$1 \; - \; \sum_{i \in U} x'_{i,j} \; - \; r'_j \;\; = \;\; \sum_{i \in I}x^*_{i,j} \; + \; \sum_{i \in G} x^{(\text{II})}_{i,j} \qquad \text{and} \quad t'_j = 1.$$
		%$t'_j \; = \; \left( \; 1 - \sum_{i \in U} x'_{i,j} - r'_j \; \right) / \sum_{i \in I} x^*_{i,j} \; = \; \sum_{i \in I} x^*_{i,j} / \sum_{i \in I} x^*_{i,j} \; = \; 1$.
		
	\item
		If $j \in J^{(\leftrightarrow)}$ and $r'_j := \sum_{i \in U} x'_{i,j}$, 
		%
		then by the conclusion of Lemma~\ref{lemma-assignment-absorbing-process} 
		we have $\sum_{i \in I} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} + \sum_{i \in U} x'_{i,j} > 1/2$, which implies that
		%it follows that
		$$1 \; - \; \sum_{i \in U} x'_{i,j} \; - \; r'_j \;\; = \;\; 1 \; - \; 2\cdot \sum_{i \in U} x'_{i,j} \;\; < \;\;  2 \cdot \left( \; \sum_{i \in I}x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} \; \right)$$
		%
		and $t'_j < 2$.
		% / \sum_{i \in I} x^*_{i,j} \; < \; \left(\; 2\sum_{i \in I} x^*_{i,j} \; \right) / \sum_{i \in I} x^*_{i,j} \; = \; 2.$
\end{itemize}
%
%\noindent
In all cases we have $t'_\ell \le 2$.
%
\end{proof}

%

%\medskip

%
%



\begin{figure*}[h]
\centering
\fbox{
\begin{minipage}{.8\textwidth}
\begin{align}
& \text{min} & & \sum_{i \in G} \; y_i \; + \; \sum_{i \in G, \hspace{1pt} j \in U} c_{i,j} \cdot x_{i,j}  & &  \label{LP-CFL-outliers} \tag*{LP-(O)} \\[3pt]
& \text{s.t.} & & \sum_{i\in G} \; x_{i,j} \; \ge \; d_j, & & \forall j\in U, \tag*{(O-1)} \label{LP-CFL-outliers-M1} \\[2pt]
& & & \sum_{j \in U} \; x_{i,j} \; \le \; u_i \cdot y_i, & & \forall i \in G, \tag*{(O-2)} \label{LP-CFL-outliers-M2} \\[3pt]
& & & y_i \; \le \; 1, & & \forall i \in G, \tag*{(O-3)} \label{LP-CFL-outliers-M3} \\[4pt]
& & & x_{i,j} \; \ge \; 0, \;\; y_i \; \ge \; 0, & & \forall i \in G, \; j \in U. \tag*{(O-4)} \label{LP-CFL-outliers-M4}
\end{align}
\vspace{-10pt}
\end{minipage}\quad
}
\caption{(Restate for further reference) The assignment LP for clusters centered at outlier clients.}
\label{fig-LP-CFL-outliers}
\end{figure*}



%%%
%%%

\paragraph{The bundled assignment $\bm{g}$ for~\ref{LP-CFL-outliers}.}


In the following we consider the assignment LP for rounding the clusters in $\MC{C}_{H'}$.
%\mong{*** polish below ******}
%
%Consider the rounding process for clusters in $\MC{C}_{H'}$ in the second phase.
%
For any $w \in U$ and $i \in G$ such that $i \in B(k)$ for some $k \in H(w)$, i.e., $i$ belongs to the clusters centered at some $k \in H(w)$, define the bundled assignment $g_{i,w}$ as
$$g_{i,w} \; := \; \sum_{\ell \in \MC{D} \cup H} t'_\ell \cdot x'^{(\text{II})}_{i,\ell}. $$
%
The following lemma shows that the feasible region of~\ref{LP-CFL-outliers} is nonempty, and the basic optimal solution $(\bm{x}'', \bm{y}'')$ exists.
%

%The following lemma, which shows the feasibility
% of $\left( \; \left.\bm{g}\right|_{G,U}, \; 2 \left.\bm{y}'^{(\text{II})}\right|_{G} \; \right)$ 
%for~\ref{LP-CFL-outliers}, is proved by verifying the corresponding constraints.
% of~\ref{LP-CFL-outliers}.

\begin{lemma} \label{lemma-assignment-lp-feasibility}
%
%Furthermore, 
$\left( \; \left.\bm{g}\right|_{G,U}, \; 2\left.\bm{y}'^{(\text{II})}\right|_{G} \; \right)$ is a feasible solution for~\ref{LP-CFL-outliers}.
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-assignment-lp-feasibility}]
%
We prove by verifying the constraints of~\ref{LP-CFL-outliers}.
% is straightforward.
%
\begin{itemize}
	\item
		Consider~\ref{LP-CFL-outliers-M1} in~\ref{LP-CFL-outliers}. 
		
		For any $w \in U$, apply the definition of $\bm{g}$ and the definition of $d_w$, we have 
		\begin{align*}
		\sum_{i \in G} \; g_{i,w} 
		%\; = \; \sum_{i \in G \colon i \in B(k) \text{ for some } k \in H(w)} \; g_{i,w} 
		\; = \; \sum_{ k \in H(w), \; i' \in B(k)} \; \sum_{\ell \in \MC{D} \cup H} \; t'_\ell \cdot x'^{(\text{II})}_{i',\ell} \; = \; d_w \; \ge \; d_w.
		\end{align*}
		
	\item
		Consider~\ref{LP-CFL-outliers-M3} in~\ref{LP-CFL-outliers}.
		
		For any $i \in G$, since $G \subseteq I$, it follows that $y'^{(\text{II})}_i \le y'_i \le \; 1/2$, and we have $2\cdot y'^{(\text{II})}_i \; \le \; 1$.
		
	\item
		Consider~\ref{LP-CFL-outliers-M2} in~\ref{LP-CFL-outliers}.
		
		For any $i \in G$, let $k \in H$ be the outlier client such that $i \in B(k)$. 
		%
		Applying the definition of $\bm{g}$ and 
		%the fact that $t'_j \le 2$ for all $j \in \MC{D} \cup H$, 
		Corollary~\ref{cor-scaling-factor}, we have 
		\begin{align*}
		\sum_{w \in U} g_{i,w} \; 
		& = \; g_{i,w(k)} \; = \; \sum_{\ell \in \MC{D} \cup H} t'_\ell \cdot x'^{(\text{II})}_{i,\ell} \; \le \; \sum_{\ell \in \MC{D} \cup H} 2\cdot x'^{(\text{II})}_{i,\ell} \; \le \; 2\cdot u_i \cdot y'^{(\text{II})}_i,
		\end{align*}
		where in the last inequality we apply the conclusion of Corollary~\ref{cor-feasibility-cons} which states that constraint~\ref{cons-P-2-with-H} holds for $i$ when $i$ is removed from $F'$ in the first phase.
		% of the rounding process.
		% Lemma~\ref{lemma-feasibility-absorbing-process} and use the fact that $\left( \left.\bm{x}'^{(\text{II})}\right|_{G,\MC{D}\cup H}, \left.\bm{y}'^{(\text{II})}\right|_G \right)$ satisfies the constraint~\ref{cons-P-2} in~\ref{LP-natural-CFL}.
		% \mong{****}
		%
\end{itemize}
%
This proves the lemma.
\end{proof}

%

%%%
%%%

\paragraph{The unbundled assignment $\bm{h}$ from $\bm{x}''$.}

For each $i \in G$ and $j \in \MC{D} \cup H$, define the unbundled assignment $h_{i,j}$ from $\bm{x}''$ as
%\mong{*** incorrect ***}
$$h_{i,j} \hspace{4pt} := \hspace{4pt} \sum_{w \in U} \; x''_{i,w} \cdot \frac{1}{d_w} \cdot \sum_{k \in H(w), \; i' \in B(k)} t'_j \cdot x'^{(\text{II})}_{i',j}.$$
%where $w \in U$ is the facility at which $j(q)$ is located.
%
Intuitively, in $h$ we redistribute the bundled assignment $\bm{x}''$ back for the original clients, and it follows that for any $j \in \MC{D} \cup H$, 
\begin{align}
\sum_{i \in G} \; h_{i,j} \;\; 
& = \;\; \sum_{ i \in G, \; w \in U} \; x''_{i,w} \cdot \frac{1}{d_w} \cdot \sum_{k \in H(w), \; i' \in B(k)} t'_j \cdot x'^{(\text{II})}_{i',j}  \notag \\[6pt]
%& = \; \sum_{ w \in U } \; \sum_{i \in G} x''_{i,w} \cdot \frac{1}{d_w} \cdot \sum_{k \in H(w)}\; \sum_{i' \in B(k)} t'_j \cdot x'^{(\text{II})}_{i',j} \\
& = \;\; \sum_{w \in U} \; \sum_{k \in H(w), \; i' \in B(k)} t'_j \cdot x'^{(\text{II})}_{i',j} 
\;\; = \;\; \sum_{i \in G} t'_j \cdot x'^{(\text{II})}_{i,j},
\label{ieq-unbundle-fully-assigned}
\end{align}
where in the second equality we apply~\ref{LP-CFL-outliers-M1} from~\ref{LP-CFL-outliers} for the fact that $\sum_{i \in G} x''_{i,w} = d_w$ for all $w \in U$ since $(\bm{x}'', \bm{y}'')$ is a feasible solution for~\ref{LP-CFL-outliers}
% assigns $d_w$ amount of demand to $G$ for all $w \in U$ 
and in the last equality we use the fact that the set of satellite facilities for each $j \in H$ forms a partition of $G$.
%
%Note that, 
%For each $j \in H$, define the assignment 

%\smallskip

%
%For each $q \in \MC{C}_{D'}$ and each $j \in \MC{D} \cup H$, define the absorbed assignment 
%$$h^{(q)}_{i(q),j} \; := \; x'^{(q)}_{i(q),j} \; + \; \delta_{i(q)} \cdot \sum_{\ell \in B(q)} x'^{(q)}_{\ell,j}.$$
%
%Intuitively, $h^{(q)}$ consists of the assignments that was made to facility $i(q)$ and the assignments for which $i(q)$ has absorbed from the satellite facilities. 
%in $N_{(I \cap F',x')}(j(q))$ in the iteration when $q$ is formed.
%
%
%\smallskip
%
%
%Similarly, for each $q \in \MC{C}_{H'}$, each $i \in B(j(q))$, and each $j \in \MC{D}$, define
%$$h^{(q)}_{i,j} \; := \; \frac{x'_{i,j}}{d_w} \cdot x''_{i,w},$$
%where $w \in U$ is the facility at which $j(q)$ is located.
%
%Intuitively, in $h^{(q)}$ we redistribute the rounded assignment back to the original clients.
%
%For each $j \in \MC{D}$, define the scaling factor $t'_j$ as %to be
%$$t'_j := \frac{1}{\sum_{q \in \MC{C}_{D'}} h^{(q)}_{i(q),j} } \cdot \left( \; 1 - \sum_{i \in U} x'_{i,j} - r'_j \; \right) \quad\enskip \text{if $\sum_{q \in \MC{C}_{D'}} h^{(q)}_{i(q),j} > 0,$}$$
%and $t'_j := 1$ otherwise.
%
%For the ease of presentation, for any $j \in H$, also define $t'_j := 1$.
%

\medskip

%%%
%%%

\paragraph{The Assignment $\bm{x}^\circ$.}
\label{par-proof-approx-cfl-cfc-feasibility-xc}

%

%
Provided above, the assignment $\bm{x}^\circ$ made for each $j \in \MC{D}$ is defined as
$$\left.\bm{x}^\circ \right|_{j} \; := \; 
\left.\bm{x}'\right|_{U,\{j\}} \; + \; t'_j \cdot \left. \bm{x}^*\vphantom{^{(q)}} \right|_{F^*_{D'}, \{j\}} \; + \; \left. \bm{x}^*\vphantom{^{(q)}} \right|_{F^*_{D'},H(j)} \; + \; \left. \bm{h}\vphantom{^{(q)}} \right|_{G, \{j\} \cup H(j)}.$$
%$$\left.\bm{x}^\circ \right|_{j} \; := \; 
%\left.\bm{x}'\right|_{U,\{j\}} \; + \; \sum_{q \in \MC{C}_{D'}} \left( \;\; t'_j \cdot \left. \bm{h}^{(q)} \right|_{\{j\}} \; + \; \left. \bm{h}^{(q)} \right|_{H(j)} \;\; \right) \; + \; \left. \bm{h}\vphantom{^{(q)}} \right|_{\{j\} \cup H(j)},$$
%where for any assignment $\bm{x}$ we use $\left.\bm{x}\right|_{A,B}$ to denote the assignment made in $\bm{x}$ between $A \subseteq \MC{F}$ and $B \subseteq \MC{D}$.
% and $\left.\bm{x}\right|_{B}$ to denote the assignment made in $\bm{x}$ for $B \subseteq \MC{D}$, respectively.
%
% $\left.\bm{x}'\right|_{U,j}$ denotes the assignments made in $\bm{x}'$ between $U$ and $j$, and $\left. h^{(q)} \right|_{j}$, $\left. h \right|_{q}$ denote the assignments made in $h^{(q)}$ and $h$ for the client $j$ and $q$, respectively.
%
%
%\smallskip
%
More precisely, for any $i \in \MC{F}$, $j \in \MC{D}$,
$$x^\circ_{i,j} \;\; := \;\; \begin{cases}
\; x'_{i,j}, & \text{if $i \in U$,} \\[3pt]
\; t'_j \cdot x^*_{i,j} \; + \; \sum_{k \in H(j)} x^*_{i,k}, \enskip & \text{if $i \in F^*_{D'}$,} \\[3pt]
\; h_{i,j} \; + \; \sum_{k \in H(j)} h_{i,k}, & \text{if $i \in G$,} \\[2pt]
\; 0, & \text{otherwise}.
\end{cases}$$
%
Intuitively, the assignment of each $j \in \MC{D}$ in $\bm{x}^\circ$ consists of its original assignments to $U$ and the rounded assignments for clients in $\{j\} \cup H(j)$ to facilities in $F^*_{D'} \cup G$.
% in $\MC{C}_{D'} \cup \MC{C}_{H'}$.

\smallskip

%
The following lemma, which asserts the feasibility of $\bm{x}^\circ$, is 
%therefore 
straightforward to verify. 
%

\begin{lemma} \label{lemma-cfl-ufc-feasibility}
%We have $t'_j \le 2$ for all $j \in \MC{D}$.
%
%Furthermore, 
%the solution 
$(\bm{x}^\circ, \bm{y}^*)$ is feasible for~\ref{LP-natural-CFL} on the input instance $\Psi$.
% = (\MC{F},\MC{D}, \MB{c}, \MB{u})$.
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cfl-ufc-feasibility}]
%
In the following we show that $(\bm{x}^\circ, \bm{y}^*)$ is feasible for~\ref{LP-natural-CFL} on $\Psi$.
%
Since $\bm{y^*}$ is already integral and takes values only from $\{0, 1\}$, it suffices to argue that $\bm{x}^\circ$ fully-assigns each $j \in \MC{D}$ and respects the capacity constraints given by $\bm{y}^*$.
%

\smallskip

%
For the latter part, since $\bm{x}^\circ$ keeps the assignments of $\MC{D}$ to $U$ unchanged, it suffices to argue for the assignments to $F^*_{D'} \cup G$.
%
By the definition of $\bm{x}^\circ$, Corollary~\ref{cor-scaling-factor}, and Lemma~\ref{lemma-feasibility-absorbing-process}, 
%Lemma~\ref{lemma-assignment-lp-feasibility}, and , it follows that, 
for any $i \in F^*_{D'}$, we have $$\sum_{j \in \MC{D}} x^\circ_{i,j} \; = \; \sum_{j \in \MC{D}} t'_j \cdot x^*_{i,j} \; + \; \sum_{j \in H} x^*_{i,j} \; \le \; \sum_{j \in \MC{D} \cup H} 2\cdot x^*_{i,j} \; \le \; u_i \; = \; u_i \cdot y^*_i.$$
%for any $i \in F^*_{D'}$.
%
Similarly, for any $i \in G$, applying the definition of $\bm{x}^\circ$ and $\bm{h}$, and the fact that $(\bm{x}'', \bm{y}'')$ is feasible for~\ref{LP-CFL-outliers}, we have 
\begin{align*}
\sum_{j \in \MC{D}} \; x^\circ_{i,j} \; 
& = \; \sum_{j \in \MC{D} \cup H} h_{i,j} \; = \; \sum_{j \in \MC{D} \cup H} \; \sum_{w \in U} x''_{i,w} \cdot \frac{1}{d_w} \cdot \sum_{k \in H(w), \; i' \in B(k)} t'_j \cdot x'^{(\text{II})}_{i',j} \\[4pt]
& = \; \sum_{w \in U} \; x''_{i,w} \cdot \frac{1}{d_w} \cdot \sum_{k \in H(w), \; i' \in B(k)} \; \sum_{j \in \MC{D} \cup H} t'_j \cdot x'^{(\text{II})}_{i',j}
\; = \; \sum_{w \in U} x''_{i,w} \; \le \; u_i \cdot y''_i,
\end{align*}
where in the second last equality we apply the definition of $d_w$ for each $w \in U$.
%

\smallskip

%
In the following, we show that $\bm{x}^\circ$ fully-assigns each $j \in \MC{D}$.
%
It suffices to argue for the clients in $J^{(I)} \cup J^{(\leftrightarrow)}$.
%
For the former case, for any $j \in J^{(I)}$, we have $\sum_{i \in U} x'_{i,j} = 0$ and $H(j) = \emptyset$.
%
Hence,
\begin{align*}
\sum_{i \in \MC{F}} x^\circ_{i,j} \;
= & \; \sum_{i \in F^*_{D'}} t'_j \cdot x^*_{i,j} \; + \; \sum_{i \in G} \; h_{i,j} \; \\
= & \; \sum_{i \in F^*_{D'}} t'_j \cdot x^*_{i,j} \; + \; \sum_{i \in G} \; t'_j \cdot x'^{(\text{II})}_{i,j} \; = \; t'_j \cdot \left( \; \sum_{i \in F^*_{D'}} x^*_{i,j} \; + \; \sum_{i \in G} x'^{(\text{II})}_{i,j} \; \right) \; = \; 1,
\end{align*}
where 
%in the first equality we use the fact that $\sum_{i \in U} x'_{i,j} = 0$ and $H(j) = \emptyset$, 
in the second equality we apply Equality~(\ref{ieq-unbundle-fully-assigned}), and in the last equality we apply the definition of $t'_j$ with the fact that $\sum_{i \in U} x'_{i,j} = r'_j = 0$.
%

\smallskip

%
For $j \in J^{(\leftrightarrow)}$, we have
\begin{align}
\sum_{i \in \MC{F}} \; x^\circ_{i,j} \;\; = \;\; \sum_{i \in U} \; x'_{i,j} \; + \; \sum_{i \in F^*_{D'}} \left( \; t'_j \cdot x^*_{i,j} \; + \; \sum_{k \in H(j)} x^*_{i,k} \; \right) \; + \; \sum_{i \in G} \left( \; h_{i,j} \; + \; \sum_{k \in H(j)} h_{i,k} \; \right)
\label{ieq-j-lr-fully-assign-1}
\end{align}
%
Applying Equality~(\ref{ieq-unbundle-fully-assigned}) and the definition of $t'_k$ for $k \in H(j)$, we have
\begin{align}
\sum_{i \in G} \left( \; h_{i,j} \; + \; \sum_{k \in H(j)} h_{i,k} \; \right) \;\; = \;\; \sum_{i \in G} \; t'_j \cdot x'^{(\text{II})}_{i,j} \; + \; \sum_{k \in H(j), \; i \in G} x'^{(\text{II})}_{i,k}
\label{ieq-j-lr-fully-assign-2}
\end{align}
%
Combining~(\ref{ieq-j-lr-fully-assign-1}) and~(\ref{ieq-j-lr-fully-assign-2}), we have
\begin{align}
\sum_{i \in \MC{F}} \; x^\circ_{i,j} \;\; 
& = \;\; \sum_{i \in U} x'_{i,j} \; + \; \sum_{i \in F^*_{D'}} t'_j \cdot x^*_{i,j} \; + \; \sum_{i \in G} t'_j \cdot x'^{(\text{II})}_{i,j} \; + \; \sum_{k \in H(j)} \left( \; \sum_{i \in F^*_{D'}} x^*_{i,k} \; + \; \sum_{i \in G} x'^{(\text{II})}_{i,k} \; \right) \notag \\[6pt]
& = \;\; \sum_{i \in U} x'_{i,j} \; + \; \sum_{i \in F^*_{D'}} t'_j \cdot x^*_{i,j} \; + \; \sum_{i \in G} t'_j \cdot x'^{(\text{II})}_{i,j} \; + \; \sum_{k \in H(j)} \; \sum_{i \in I} x'_{i,k},
\label{ieq-j-lr-fully-assign-3}
\end{align}
where in the last equality we apply the conclusion of Lemma~\ref{lemma-assignment-absorbing-process}.
%
By the construction of outlier clients, each $k \in H(j)$ is fully-assigned to facilities in $I$ by $\bm{x}'$.
%
Hence, we have $\sum_{i \in I} x'_{i,k} = d_k$ for each $k \in H(j)$.
%
Further applying the definition of $d_k$, 
%
we have
\begin{align}
\sum_{k \in H(j)} \; \sum_{i \in I} x'_{i,k} \; = \; \sum_{k \in H(j)} d_k \; = \; \sum_{k \in H(j)} r'_j \cdot \frac{x'_{w(k),j}}{\sum_{i \in U} x'_{i,j}} \; = \; r'_j,
\label{ieq-j-lr-fully-assign-4}
\end{align}
where the last equality follows from the fact that exactly one outlier client is created for each $i \in U$ with $x'_{i,j} > 0$.
%
Combining~(\ref{ieq-j-lr-fully-assign-3}) and~(\ref{ieq-j-lr-fully-assign-4}) and applying the definition of $t'_j$, we have
$$\sum_{i \in \MC{F}} \; x^\circ_{i,j} \;\; 
= \;\; \sum_{i \in U} x'_{i,j} \; + \; \sum_{i \in F^*_{D'}} t'_j \cdot x^*_{i,j} \; + \; \sum_{i \in G} t'_j \cdot x'^{(\text{II})}_{i,j} \; + \; r'_j \;\; = \;\; 1.$$
%
This completes the proof of this lemma.
\end{proof}

%

%

%

%%%
%%%

%\paragraph{Approximation Guarantee.}
\subsection{Approximation Guarantee}
\label{subsec-proof-approx-cfl-cfc-guarantee}

%

%\smallskip

%
In the following we establish our approximation guarantee for CFL-CFC.
%
%While bounding the facility cost is rather straightforward, w
We will bound the cost incurred by clusters in $\MC{C}_{H'}$ and $\MC{C}_{D'}$ separately.
%We show that $\Psi(\bm{x}^\circ, \bm{y}^*) \le 4 \cdot \Psi(\bm{x}', \bm{y}')$ by considering the cost incurred by the clusters formed during the process.
%
First we bound below the cost incurred by $\MC{C}_{H'}$.
%
We bound the cost incurred by $\MC{C}_{D'}$ in page~\pageref{par-proof-approx-cfl-cfc-cost-cd} and establish the overall guarantee in page~\pageref{par-proof-approx-cfl-cfc-overall-cost}.
%
%\mong{**** recall the definition of $\bm{x}^\circ$ ****}

\smallskip

Recall that we use $p(j)$ for each $j \in H$ to denote the client in $\MC{D}$ from which $j$ is created.
%
In the following, we extend this definition and define $p(k) := k$ for each $k \in \MC{D}$ for notational brevity.
%\mong{**** move to the beginning? *****}

%

%%%
%%%

\paragraph{The Clusters in $\MC{C}_{H'}$.}
\label{par-proof-approx-cfl-cfc-cost-ch}

%
Consider the cost incurred by clusters in $\MC{C}_{H'}$.
%
We begin with the following lemma regarding the assignment radius of the outlier clients in $H$.
%


\begin{lemma} \label{lemma-bound-alpha-dist}
For any $j \in H$ and $i \in G$ such that $x'^{(\text{II})}_{i,j} > 0$, we have $c_{i,j} \; \le \; \alpha_j$.
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-bound-alpha-dist}]
%
By the algorithm design, the outlier client $j$ is created and assigned to $i$ in $\bm{x}'$ only when $x'_{i,p(j)} > 0$ holds in the beginning of the rounding process, which implies that $c_{i,p(j)} \; \le \; \alpha_{p(j)}$ by complementary slackness condition.
%
By triangle inequality and the definition of $\alpha_j$, it follows that 
$c_{i,j} \; = \; c_{i,w(j)} \; \le \; c_{w(j),p(j)} \; + \; c_{i,p(j)} \; \le \; c_{w(j),p(j)} \; + \; \alpha_{p(j)} \; = \; \alpha_j$.
%
\end{proof}

%
The following lemma, which bounds the assignment cost of $\left.\bm{x}^\circ\right|_{G,\MC{D}}$ in terms of that in $\bm{x}''$ and $\left.\bm{x}'^{(\text{II})}\right|_{G,\MC{D}\cup H}$, follows from the way the clusters in $\MC{C}_{H'}$ are formed and the assignments are bundled.
%within clusters in $\MC{C}_{H'}$ are bundled.
%outlier clients are created,
%via rerouting the assignments accordingly, 
%.
%

\begin{lemma} \label{lemma-bound-outlier-lp-rerouting-cost}
%
%\mong{**** double check **** $c_{i,p(j)}$ ****}
$$\sum_{i \in G, \; j \in \MC{D}} c_{i,j} \cdot x^\circ_{i,j} \; \le \; \sum_{i \in G, \; j \in U} c_{i,j} \cdot x''_{i,j} \; + \; \sum_{i \in G} \; \sum_{j \in \MC{D} \cup H} t'_j \cdot \left( \; c_{i,p(j)} + \alpha_j \; \right) \cdot x'^{(\text{II})}_{i,j}.$$
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-bound-outlier-lp-rerouting-cost}]
%
%\mong{*** recall notations defined far away ***}
%

\smallskip

%
By the definition of $\bm{x}^\circ$ and $\bm{h}$, we have 
\begin{align}
\sum_{i \in G, \; j \in \MC{D}} c_{i,j} \cdot x^\circ_{i,j} \;\; 
& = \;\; \sum_{i \in G, \; j \in \MC{D}} \; \sum_{k \in \{j\} \cup H(j)} c_{i,j} \cdot h_{i,k} \;\; = \;\; \sum_{i \in G} \; \sum_{j \in \MC{D} \cup H} c_{i,p(j)} \cdot h_{i,j}  \notag \\[6pt]
& = \;\; \sum_{i \in G} \; \sum_{j \in \MC{D} \cup H} \; c_{i,p(j)} \cdot \sum_{w \in U} \; x''_{i,w} \cdot \frac{1}{d_w} \cdot \sum_{k \in H(w), \; i' \in B(k)} t'_j \cdot x'^{(\text{II})}_{i',j}.
\label{ieq-bound-outlier-lp-rerouting-cost-1}
\end{align}
%
By triangle inequality, for any $i \in G$, $j \in \MC{D} \cup H$, $w \in U$, $k \in H(w)$, and $i' \in B(k)$ such that $x'^{(\text{II})}_{i',j} > 0$, we have
$$c_{i,p(j)} \; \le \; c_{i,w} \; + \; c_{i',w} \; + \; c_{i',p(j)} \; \le \; c_{i,w} \; + \; \alpha_k \; + \; c_{i',p(j)},$$
where the last inequality follows from Lemma~\ref{lemma-bound-alpha-dist} and the fact that $i' \in B(k)$ implies that $x'^{(\text{II})}_{i',k} > 0$.
%
See also Figure~\ref{fig-outlier-client-assignment} for an illustration.
%
%Consider $\vphantom{\text{\Large T}} c_{i',p(j)}$.
%
%If $j \in \MC{D}$, then we have $c_{i',p(j)} = c_{i',j}$ by definition.
%
%If $j \in H$, then the assumption that $x'^{(\text{II})}_{i',j} > 0$ implies that $c_{i',p(j)} \le c_{i',j}$ by Lemma~\ref{lemma-bound-alpha-dist}.
%
%Hence, we have $c_{i,p(j)} \; \le \; c_{i,w} \; + \; \alpha_k \; + \; c_{i',j}$.
%

\begin{figure*}[t]
\centering
\fbox{
\begin{minipage}{.3\textwidth}
\centering
\includegraphics[scale=1]{outlier-client-assignment}
\end{minipage}
}
\caption{An illustration on the bundled assignment from $w\in U$ to $i \in G$ and unbundled assignments for $k \in H(w)$, $i' \in B(k)$ such that $x'^{(\text{II})}_{i',j} > 0$.}
\label{fig-outlier-client-assignment}
\end{figure*}

\smallskip

%
Applying the above inequality on~(\ref{ieq-bound-outlier-lp-rerouting-cost-1}) with proper rearrangement, we obtain
\begin{align*}
& \sum_{i \in G, \; w \in U} c_{i,w} \cdot x''_{i,w} \cdot \frac{1}{d_w} \cdot \sum_{k \in H(w), \; i' \in B(k)} \; \sum_{j \in \MC{D}\cup H} t'_j \cdot x'^{(\text{II})}_{i',j} \\[4pt]
& \qquad\quad + \;\; \sum_{i \in G, \; w \in U} \; x''_{i,w} \cdot \frac{1}{d_w} \cdot \sum_{k \in H(w), \; i' \in B(k)} \; \sum_{j \in \MC{D} \cup H} \left( \; c_{i',p(j)} + \alpha_k \; \right) \cdot t'_j \cdot x'^{(\text{II})}_{i',j}.
\end{align*}
%
Applying the definition of $d_w$ on the former term and the fact that $\sum_{i \in G} x''_{i,w} = d_w$ by the fact that $(\bm{x}'',\bm{y}'')$ is a feasible solution for~\ref{LP-CFL-outliers} on the latter term, the above becomes
%
\begin{align*}
\sum_{i \in G, \; w \in U} c_{i,w} \cdot x''_{i,w} \; + \; \sum_{w \in U} \; \sum_{k \in H(w), \; i' \in B(k)} \; \sum_{j \in \MC{D} \cup H} \left( \; c_{i',p(j)} + \alpha_k \; \right) \cdot t'_j \cdot x'^{(\text{II})}_{i',j}. 
\end{align*}
%
By the design of the algorithm, for any $w \in U, \; k \in H(w), \; i' \in B(k)$, and for any $j \in \MC{D} \cup H$ with $x'^{(\text{II})}_{i',j} > 0$, we have $\alpha_k \le \alpha_j$, since $k$ is selected as cluster center because of having the lowest $\alpha$ value.
%
Therefore, the above is further bounded by
\begin{align*}
& \sum_{i \in G, \; w \in U} c_{i,w} \cdot x''_{i,w} \; + \; \sum_{w \in U} \; \sum_{k \in H(w), \; i' \in B(k)} \; \sum_{j \in \MC{D} \cup H} \left( \; c_{i',p(j)} + \alpha_j \; \right) \cdot t'_j \cdot x'^{(\text{II})}_{i',j}.
\end{align*}
Applying the fact that the satellite facilities of clusters in $\MC{C}_{H'}$ forms a partition of $G$, the above is exactly
\begin{align*}
\sum_{i \in G, \; j \in U} c_{i,j} \cdot x''_{i,j} \; + \; \sum_{i \in G} \; \sum_{j \in \MC{D} \cup H} t'_j \cdot \left( \; c_{i,p(j)} + \alpha_j \; \right) \cdot x'^{(\text{II})}_{i,j}.
\end{align*}
%where in the third equality we apply the definition of $\bm{h}$, in the fourth inequality we use the fact that $c_{i,p(j)} \le c_{i,w} + c_{i',j} + \alpha_k$, and in the fifth inequality we use the fact that, the center client of any cluster has the lowest $\alpha$ value, i.e., for the cluster centered at an outlier client $k \in H$, any $j \in \MC{D} \cup H$ with $x'^{(\text{II})}_{i',j} > 0$ for some $i' \in B(k)$ has $\alpha_j \ge \alpha_k$.
%
%\mong{****}
%
\end{proof}

%
%First we bound the assignment cost incurred by $\bm{x}''$.
%
%\mong{**** fix this **** $\bm{x}'^{(\text{II})}$ needs to be scaled up ****}
%
%Let 
%
The following lemma
%, which follows from Lemma~\ref{lemma-assignment-lp-feasibility} and the fact that $(\bm{x}'', \bm{y}'')$ is a basic optimal solution, 
bounds the cost incurred by $\left( \bm{x}'', \bm{y}'' \right)$.
%

\begin{lemma} \label{lemma-bound-outlier-lp-assignment-cost}
%
We have
$$\sum_{i \in G} \left\lceil y''_i \right\rceil \; + \sum_{i \in G, \; j \in U} c_{i,j} \cdot x''_{i,j} \hspace{4pt} \le \hspace{4pt} 2 \cdot \sum_{i \in G}y'^{(\text{II})}_i \; + \; |L| \; + \sum_{i \in G} \; \sum_{j \in \MC{D} \cup H} t'_j \cdot \alpha_j \cdot x'^{(\text{II})}_{i,j} ,$$
where $L := \left\{ \; \vphantom{\text{\Large T}} i \in G \; \colon \; 0 < y''_i < 1 \; \right\}$.
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-bound-outlier-lp-assignment-cost}]
%
Since $(\bm{x}'', \bm{y}'')$ is optimal for~\ref{LP-CFL-outliers}, by Lemma~\ref{lemma-assignment-lp-feasibility}, the cost of $(\bm{x}'', \bm{y}'')$ is no more than that of $\left( \; \left.\bm{g}\right|_{G,U} \; , \; 2\left.\bm{y}'\right|_G \; \right)$.
%
Hence,
\begin{align*}
\sum_{i \in G} y''_i \; + \sum_{i \in G, \; j \in U} c_{i,j} \cdot x''_{i,j} \hspace{4pt} 
& \le \hspace{4pt} 2 \cdot \sum_{i \in G}y'^{(\text{II})}_i \; + \; \sum_{i \in G, \; j \in U} \; c_{i,j} \cdot g_{i,j} \\[6pt]
& = \hspace{4pt} 2 \cdot \sum_{i \in G}y'^{(\text{II})}_i \; + \; \sum_{\substack{i \in G, \; w \in U, \\[3pt] i \in B(k) \text{ for some } k \in H(w)}} \; c_{i,w} \cdot \sum_{\ell \in \MC{D} \cup H} t'_\ell \cdot x'^{(\text{II})}_{i,\ell}.
\end{align*}
%
By the algorithm setting and Lemma~\ref{lemma-bound-alpha-dist}, for any $i \in G$ and $w \in U$ such that $i \in B(k)$ for some $k \in H(w)$, i.e., facility $i$ belongs to some cluster centered at some $k \in H(w)$, we have $c_{i,w} \le \alpha_k \le \alpha_\ell$, for any $\ell \in \MC{D} \cup H$ with $x'^{(\text{II})}_{i,\ell} > 0$.
%
Therefore,
\begin{align*}
\sum_{i \in G} y''_i \; + \sum_{i \in G, \; j \in U} c_{i,j} \cdot x''_{i,j} \hspace{4pt} 
& \le \hspace{4pt} 2 \cdot \sum_{i \in G}y'^{(\text{II})}_i \; + \; \sum_{\substack{i \in G, \; w \in U, \\[3pt] i \in B(k) \text{ for some } k \in H(w)}} \; \sum_{\ell \in \MC{D} \cup H} t'_\ell \cdot \alpha_\ell \cdot x'^{(\text{II})}_{i,\ell} \\[6pt]
& = \hspace{4pt} 2 \cdot \sum_{i \in G}y'^{(\text{II})}_i \; + \; \sum_{i \in G, \; j \in \MC{D} \cup H} t'_j \cdot \alpha_j \cdot x'^{(\text{II})}_{i,j},
\end{align*}
where the last equality follows from the fact that the set of satellite facilities of clusters in $\MC{C}_{H'}$ forms a partition of $G$.
%
Applying the definition of $L$ competes the proof of this lemma.
%
\end{proof}

%
The following lemma follows from the fact that $(\bm{x}'', \bm{y}'')$ is a basic solution for~\ref{LP-CFL-outliers}.
%

\begin{lemma} \label{lemma-bound-outlier-lp-basic-solution}
%
%We have 
%$|L| \le |U|$, where $L := \left\{ \; \vphantom{\text{\Large T}} i \in G \; \colon \; 0 < y''_i < 1 \; \right\}$.
$$|L| \; \le \; |U|, \quad \text{where } L := \left\{ \; \vphantom{\text{\LARGE T}} i \in G \; \colon \; 0 < y''_i < 1 \; \right\}.$$
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-bound-outlier-lp-basic-solution}]
%
%We will show that the fact that $(\bm{x}'', \bm{y}'')$ is a basic feasible solution for~\ref{LP-CFL-outliers} implies the above.
%
%\medskip
%
Consider the set of constraints in~\ref{LP-CFL-outliers} that hold with equality at $(\bm{x}'', \bm{y}'')$, for which we denote by
%
$\MC{E}^{(=)}$ in the following.
%
Let $M_3$ and $M_4$ denote the set of constraints in $\MC{E}^{(=)}$ of the types~\ref{LP-CFL-outliers-M3} and~\ref{LP-CFL-outliers-M4}, respectively.
%
Formally,
$$M_3 := \BIGBP{ \vphantom{\dfrac{}{}}\hspace{3pt} i \hspace{2pt} \colon \hspace{2pt} \fbox{$\vphantom{{\dfrac{}{}}^{\bigcup}}$ $y_i \le 1$ } \in \MC{E}^{(=)} \hspace{3pt} }$$
and
$$M_4 := \BIGBP{ \vphantom{\dfrac{}{}}\hspace{3pt} (i,j) \hspace{2pt} \colon \hspace{2pt} \fbox{$\vphantom{{\dfrac{}{}}^{\bigcup}}$ $x_{i,j} \ge 0$ } \in \MC{E}^{(=)} \hspace{3pt} } \cup \BIGBP{ \vphantom{\dfrac{}{}}\hspace{3pt} i \hspace{2pt} \colon \hspace{2pt} \fbox{$\vphantom{{\dfrac{}{}}^{\bigcup}}$ $y_i \ge 0$ } \in \MC{E}^{(=)} \hspace{3pt} }.$$
%
Let $X$ be the number of variables in~\ref{LP-CFL-outliers}.
%
Since $(\bm{x}'', \bm{y}'')$ is a basic solution for~\ref{LP-CFL-outliers}, it follows that, the coefficient matrix of $\MC{E}^{(=)}$ is of full-rank, i.e., has rank $X$.
%
Since $M_3$ and $M_4$ are linearly independent, there exists a subset $\MC{E}' \subseteq \MC{E}^{(=)}$ of linearly independent constraints such that $M_3 \cup M_4 \subseteq \MC{E}'$ and $|\MC{E}'| = X$.
%

\smallskip

%
Let $\MC{E}'' := \MC{E}' \setminus ( M_3 \cup M_4)$ and modify the constraints in $\MC{E}''$ by setting the variable $y_i$ to be $1$ for all $i \in M_3$.
%
Similarly, modify $\MC{E}''$ by setting $x_{i,j}$ to be zero for all $(i,j) \in M_4$ and $y_i$ to be zero for all $i \in M_4$.
%
By doing so, we removed equally many constraints and variables from $\MC{E}'$.
%
Since $M_3$ and $M_4$ are linearly independent, it follows that $$\text{rank}(\MC{E}'') \; = \; \text{rank}(\MC{E}') - | M_3 \cup M_4|,$$
and the coefficient matrix of $\MC{E}''$ is still of full-rank.
%

%\smallskip

%
$$\text{Let} \enskip M_1 := \BIGBP{ \vphantom{\dfrac{}{}}\hspace{3pt} j \hspace{2pt} \colon \hspace{2pt} \fbox{$\vphantom{{\dfrac{}{}}^{\bigcup}}$ $\sum_{i \in G} x_{i,j} \ge d_j$ } \in \MC{E}'' \hspace{3pt} } \enskip \text{and} \enskip M_2 := \BIGBP{ \vphantom{\dfrac{}{}}\hspace{3pt} i \hspace{2pt} \colon \hspace{2pt} \fbox{$\vphantom{{\dfrac{}{}}^{\bigcup}}$ $\sum_{j \in U} x_{i,j} \le u_i \cdot y_i$ } \in \MC{E}'' \hspace{3pt} } .$$
%
Also let $H := \left\{ \vphantom{\text{\Large T}} \; (i,j) \; \colon \; x''_{i,j} \neq 0 \; \right\}$.
%
It follows by the above setting that, $L \cup H$ corresponds exactly to the set of variables in $\MC{E}''$.
% denote the set of nonzero $x_{i,j}$ variables.
Since the coefficient matrix of $\MC{E}''$ has full rank,
%
the pivot in each row of the matrix defines a one-to-one mapping $\phi \colon L \cup H \rightarrow M_1 \cup M_2$ between the variables and the constraints.
%

\smallskip

%
Consider each $i \in L$.
%
Since the variable $y_i$ appears exactly in one constraint in $M_2$, the mapping must map $y_i$ to the constraint it corresponds to, i.e., $\phi(i) = i$.
%
Since the constraint $i$ corresponds to in $M_2$ contributes one rank, it is non-degenerated and contains at least one variable in $H$.
%
Let $x_{i,j}$ be one such variable.
%
Since $x_{i,j}$ appears in exactly two constraints, i.e., in the one $i$ corresponds to in $M_2$ and the one $j$ corresponds to in $M_1$, and since $\phi(i) = i$, it follows that $\phi\left( (i,j) \right) = j$.
%
Since the mapping $\phi$ is one-to-one, $j$ cannot be mapped to by other pairs.

\smallskip

%
Applying the above argument for each $i \in L$ results in a set consisting of distinct clients $j$ from $U$ with the same cardinality.
%
This shows that $|L| \le |U|$.
%
\end{proof}

%

\medskip

%
Applying Lemma~\ref{lemma-bound-outlier-lp-rerouting-cost}, Lemma~\ref{lemma-bound-outlier-lp-assignment-cost}, Lemma~\ref{lemma-bound-outlier-lp-basic-solution}, and the fact that $y'_i \ge 1/2$ for all $i \in U$, we obtain the following bound for the cost incurred by $\left( \left.\bm{x}^\circ\right|_{G,\MC{D}}, \left.\bm{y}^*\right|_{G} \right)$.
%
\begin{align}
\sum_{i \in G} y^*_i \; + \sum_{i \in G,\; j \in \MC{D}} c_{i,j} \cdot x^\circ_{i,j} \hspace{4pt}
& \le \hspace{4pt} 2\cdot \sum_{i \in G} y'^{(\text{II})}_i \; + \; 2 \cdot \sum_{i \in U} y'_i \; + \; \sum_{i \in G, \; j \in H} t'_j \cdot \left( \; c_{i,p(j)} + 2 \cdot \alpha_j \; \right) \cdot x'^{(\text{II})}_{i,j} \notag \\[4pt]
& \hspace{70pt} \; + \; \sum_{i \in G, \; j \in \MC{D}} t'_j \cdot \left( \; c_{i,j} + 2 \cdot \alpha_j \; \right) \cdot x'^{(\text{II})}_{i,j}  \notag \\[4pt]
& \le \hspace{4pt} 2\cdot \sum_{i \in G} y'^{(\text{II})}_i \; + \; 2 \cdot \sum_{i \in U} y'_i \; + \; \sum_{i \in G, \; j \in H} \left( \; c_{i,p(j)} + 2 \cdot \alpha_j \; \right) \cdot x'^{(\text{II})}_{i,j} \notag \\[4pt]
& \hspace{70pt} \; + \; \sum_{i \in G, \; j \in \MC{D}} \left( \; 2\cdot c_{i,j} + 2 \cdot t'_j \cdot \alpha_j \; \right) \cdot x'^{(\text{II})}_{i,j},
\label{ieq-bound-cluster-h}
\end{align}
%
where in the last inequality we apply Corollary~\ref{cor-scaling-factor} the fact that $t'_j \le 2$ for all $j \in \MC{D}$ and the definition that $t'_j = 1$ for all $j \in H$.
%, and the fact that $y'_i \ge 1/2$ for all $i \in U$, we have the following bound .

%The following lemma bounds the cost incurred by $\bm{y}''$.

%\begin{lemma} \label{lemma-bound-outlier-lp-extreme-sol}
%
%$$|L| \; \le \; |U|, \quad\enskip \text{where $L := \left\{ \; \vphantom{\text{\LARGE T}} i \in G \; \colon \; 0 < y''_i < 1 \; \right\}$.}$$
%
%\end{lemma}

%\begin{proof}[Proof of Lemma~\ref{lemma-bound-outlier-lp-extreme-sol}]
%
%\mong{*****}
%
%\end{proof}

%\smallskip

\medskip


%%%%
%%%%

\paragraph{The Clusters in $\MC{C}_{D'}$.}
\label{par-proof-approx-cfl-cfc-cost-cd}

%
In the following we consider the the clusters in $\MC{C}_{D'}$.
%
The following lemma bounds the cost incurred by each individual cluster $q$ in $\MC{C}_{D'}$.
%

\begin{lemma} \label{lemma-cost-cluster-d-prime}
%
For any $q \in \MC{C}_{D'}$, we have
%
\begin{align*}
\text{(i)} \; & & y^*_{i(q)} \; & \le \; 2 y'^{(q)}_{i(q)} \; + \; 2\delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\} } y'^{(q)}_k \enskip , \quad \text{and} \\[6pt]
\text{(ii)} \; & & \sum_{j \in \MC{D}} c_{i(q),j} \cdot x^\circ_{i(q),j} \; 
& \le \; \sum_{j \in D} 2\cdot c_{i(q),j} \cdot x'^{(q)}_{i(q),j} \; + \; \sum_{j \in H} c_{i(q),p(j)} \cdot x'^{(q)}_{i(q),j} \\[2pt]
& & & \hspace{20pt} + \; \delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\}} \; \left( \; \sum_{\; j \in \MC{D}} \; 2\cdot c_{k,j} \cdot x'^{(q)}_{k,j} \; + \; \sum_{\; j \in H} c_{k,p(j)} \cdot x'^{(q)}_{k,j} \; \right) \\[12pt]
& & & \hspace{70pt} + \; \sum_{j \in \MC{D}} \; 2 \cdot t'_j \cdot \alpha_j \cdot x^*_{i(q),j} \; + \; \sum_{j \in H} \; 2 \cdot \alpha_j \cdot x^*_{i(q),j}.
\end{align*}
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cost-cluster-d-prime}]
%
The lemma follows directly from the rounding process for clusters in $\MC{C}_{D'}$.
%
%
%\smallskip
%
%
%Let $q \in \MC{C}_{D'}$ be a cluster in $\MC{C}_{D'}$ and c
Consider the iteration for which $q$ is formed and ready to be rounded.
%
%\smallskip
%
By the algorithm design, the total facility value that has been removed from $F'$ due to the rounding process for $q$ is
\begin{align*}
y'^{(q)}_{i(q)} \; + \; \delta_{i(q)} \cdot \sum_{ k \in B(q) \setminus \{i(q)\}} y'^{(q)}_k \; = \; \frac{1}{2} \; = \; \frac{1}{2} \cdot y^*_{i(q)},
\end{align*}
where in the first equality we apply the definition of $\delta_{i(q)}$.
%
Attributing the cost of $y^*_{i(q)}$ to the facility value that is removed from $F'$ due to cluster $q$ proves the first part of this lemma.
%

\smallskip

%
For the second part, by the definition of $\bm{x}^\circ$ and the way how the algorithm reroutes the assignments from facilities in $B(q) \setminus \{i(q)\}$ to $i(q)$, we have
\begin{align}
\sum_{j \in \MC{D}} c_{i(q),j} \cdot x^\circ_{i(q),j} \;\;
= & \;\; \sum_{j \in \MC{D}} c_{i(q),j} \cdot \left( \; t'_j \cdot x^*_{i(q),j} \; + \; \sum_{\ell \in H(j)} x^*_{i(q),\ell} \; \right)  \notag \\[8pt]
= & \;\; \sum_{j \in \MC{D}} c_{i(q),j} \cdot t'_j \cdot x'^{(q)}_{i(q),j} \; + \; \sum_{\ell \in H} c_{i(q),p(\ell)} \cdot x'^{(q)}_{i(q),\ell}  \notag \\[2pt]
& \hspace{2pt} \; + \; \delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\}} \left( \; \sum_{j \in \MC{D}} c_{i(q),j} \cdot t'_j \cdot x'^{(q)}_{k,j} \; + \; \sum_{\ell \in H} c_{i(q),p(\ell)} \cdot x'^{(q)}_{k,\ell} \; \right).
\label{ieq-cost-cluster-d-prime-1}
\end{align}
%
By the algorithm setting, for any $k \in B(q) \setminus \{i(q)\}$ and any $\ell \in H$ with $x'^{(q)}_{k,\ell} > 0$, we have 
$$c_{i(q),p(\ell)} \;\; \le \;\; c_{k,p(\ell)} \; + \; c_{k,j(q)} \; + \; c_{i(q),j(q)} \;\; \le \;\; c_{k,p(\ell)} \; + \; 2\alpha_{j(q)} \;\; \le \;\; c_{k,p(\ell)} \; + \; 2\alpha_\ell,$$
%
where in the second inequality we apply the fact that $i(q)$ and $k$ are in $B(q)$, which implies that $x'^{(q)}_{i(q),j(q)} > 0$, $x'^{(q)}_{k,j(q)} > 0$, and $\max\left( c_{i(q),j(q)}, c_{k,j(q)} \right) \le \alpha_{j(q)}$ by complementary slackness, and in the last inequality we apply the assumption that $x'^{(q)}_{k,\ell} > 0$, which implies that $\ell \in H'^{(q)}$ and $\alpha_{j(q)} \le \alpha_\ell$ by the way $j(q)$ is selected.
%
%
%\mong{*******}
By a similar argument, we have $c_{i(q),j} \; \le \; c_{k,j} \; + \; 2\alpha_j$ for any $k \in B(q) \setminus \{i(q)\}$ and any $j \in \MC{D}$ with $x'^{(q)}_{k,j} > 0$.
%

\smallskip

%
By the above conclusion, Corollary~\ref{cor-scaling-factor}, and the way the assignment $\bm{x}^*$ is formed during the rounding process, 
%further applying the fact that $t'_j \le 2$ for all $j \in \MC{D}$
we have
\begin{align*}
& \delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\}, \; j \in \MC{D}} \; c_{i(q),j} \cdot t'_j \cdot x'^{(q)}_{k,j} \;\; \le \;\; \delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\}, \; j \in \MC{D}} \; \left( c_{k,j} + 2\cdot \alpha_j \right) \cdot t'_j \cdot x'^{(q)}_{k,j}  \\[8pt]
& \hspace{2.6cm} \le \;\; \delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\}, \; j \in \MC{D}} \; 2 \cdot c_{k,j} \cdot x'^{(q)}_{k,j} \; + \; \sum_{j \in \MC{D}} \;  2\cdot t'_j \cdot \alpha_j \cdot x^*_{i(q),j}.
\end{align*}
%
Similarly, we have
\begin{align*}
\delta_{i(q)} \cdot \hspace{-6pt} \sum_{k \in B(q) \setminus \{i(q)\}, \; \ell \in H} c_{i(q),p(\ell)} \cdot x'^{(q)}_{k,\ell} \;\; \le \;\; \delta_{i(q)} \cdot \hspace{-6pt} \sum_{k \in B(q) \setminus \{i(q)\}, \; \ell \in H} c_{k,p(\ell)} \cdot x'^{(q)}_{k,\ell} \; + \; \sum_{\ell \in H} \;  2\cdot \alpha_\ell \cdot x^*_{i(q),\ell}.
\end{align*}
%
Combining the above two inequalities with~(\ref{ieq-cost-cluster-d-prime-1}) and further applying Corollary~\ref{cor-scaling-factor}, we have
%
%, since $\alpha_{j(q)} \le \alpha_\ell$ for all $\ell \in D' \cup H'$ by the way $j(q)$ is selected.
%
%Further applying the fact that $t'_j \le 2$ for all $j \in \MC{D}$ and the fact that $\left.\bm{x}^*\right|_{\{i(q)\},\MC{D}\cup H}$ consists of the assignments to $i(q)$ plus the assignments $i(q)$ has absorbed, it follows that~(\ref{ieq-cost-cluster-d-prime-1}) is bounded by
%
\begin{align*}
& \sum_{j \in \MC{D}} c_{i(q),j} \cdot x^\circ_{i(q),j} \;\; \le \;\; \sum_{j \in D} \; 2\cdot c_{i(q),j} \cdot x'^{(q)}_{i(q),j} \; + \; \sum_{\ell \in H} \; c_{i(q),p(\ell)} \cdot x'^{(q)}_{i(q),\ell} \\[2pt]
& \hspace{2cm} + \; \delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\}} \; \left( \; \sum_{\; j \in \MC{D}} \; 2\cdot c_{k,j} \cdot x'^{(q)}_{k,j} \; + \; \sum_{\; \ell \in H} c_{k,p(\ell)} \cdot x'^{(q)}_{k,\ell} \; \right) \\[12pt]
& \hspace{80pt} + \; \sum_{j \in \MC{D}} \; 2 \cdot t'_j \cdot \alpha_j \cdot x^*_{i(q),j} \; + \; \sum_{\ell \in H} \; 2 \cdot \alpha_\ell \cdot x^*_{i(q),\ell}.
\end{align*}
%
\end{proof}

%

%
Take summation on the cost given in Lemma~\ref{lemma-cost-cluster-d-prime} over all clusters in $\MC{C}_{D'}$ (in the reverse order they are formed) and 
consider the contribution of each facility in $F^*_{D'} \cup G$ into the RHS of the summation.
%
%applying the fact that $F^*_{D'}$ and $G$ are mutually exclusive, w
We have the following lemma which bounds the cost incurred by $\left.\bm{x}^\circ\right|_{F^*_{D'},\MC{D}}$ and $\left.\bm{y}^*\right|_{F^*_{D'}}$.
%

\begin{lemma} \label{lemma-cost-cluster-d-prime-sum}
\begin{align}
\text{(i)} \; & & \sum_{i \in F^*_{D'}} y^*_i \;\; & \le \;\; 2\cdot \sum_{i \in I \setminus G} y'_i \; + \; 2\cdot \sum_{i \in G} \left( \; y'_i - y'^{(\text{II})}_i \hspace{1pt} \right) . \label{ieq-bound-cluster-d-p-facility} \\[8pt]
\text{(ii)} \; & & \sum_{i \in F^*_{D'},\; j \in \MC{D}} c_{i,j} \cdot x^\circ_{i,j} \;\; 
& \le \;\; \sum_{i \in F^*_{D'}, \; j \in \MC{D}} 2\cdot t'_j \cdot \alpha_j \cdot x^*_{i,j} \; + \; \sum_{i \in F^*_{D'}, \; j \in H} 2\cdot \alpha_j \cdot x^*_{i,j} \notag \\[2pt]
& & & \hspace{-60pt} \; + \; \sum_{i \in G, \; j \in \MC{D}} 2\cdot c_{i,j} \cdot \left( \; x'_{i,j} - \sum_{\ell \in H(j)} x'_{i,\ell} - x'^{(\text{II})}_{i,j} \; \right) 
\; + \; \sum_{i \in G, \; j \in H} c_{i,p(j)} \cdot \left( \; x'_{i,j} - x'^{(\text{II})}_{i,j} \; \right)  \notag \\[4pt]
& & & \hspace{-60pt} \; + \; \sum_{i \in I \setminus G, \; j \in \MC{D}} 2\cdot c_{i,j} \cdot \left( \; x'_{i,j} - \sum_{\ell \in H(j)} x'_{i,\ell} \; \right) \; + \; \sum_{i \in I \setminus G, \; j \in H} c_{i,p(j)} \cdot x'_{i,j}. \label{ieq-bound-cluster-d-p-assignment}
\end{align}
%
%\begin{align}
%\sum_{i \in F^*_{D'}} y^*_i \;\; & \le \;\; 2\cdot \sum_{i \in F^*_{D'}} y'_i \; + \; 2\cdot \sum_{i \in G} \left( \; y'_i - y'^{(\text{II})}_i \hspace{1pt} \right) \; + \; 2 \cdot \sum_{i \in I \setminus \left(F^*_{D'} \cup G\right)} y'_i \label{ieq-bound-cluster-d-p-facility} \\[8pt]
%\sum_{i \in F^*_{D'},\; j \in \MC{D}} c_{i,j} \cdot x^\circ_{i,j} \;\; 
%& \le \;\; \sum_{i \in F^*_{D'}, \; j \in \MC{D}} 2\cdot t'_j \cdot \alpha_j \cdot x^*_{i,j} \; + \; \sum_{i \in F^*_{D'}, \; j \in H} 2\cdot \alpha_j \cdot x^*_{i,j} \notag \\[8pt]
%& \hspace{20pt} \; + \; \sum_{i \in G, \; j \in H} c_{i,j} \cdot \left( \; x'_{i,j} - x'^{(\text{II})}_{i,j} \; \right) \notag \\
%& \hspace{20pt} \; + \; \sum_{i \in G, \; j \in \MC{D}} 2\cdot c_{i,j} \cdot \left( \; x'_{i,j} - \sum_{\ell \in s(j)} x'_{i,\ell} - x'^{(\text{II})}_{i,j} \; \right) \notag \\
%& \hspace{20pt} \; + \; \sum_{i \in I \setminus G, \; j \in \MC{D}} 2\cdot c_{i,j} \cdot \left( \; x'_{i,j} - \sum_{\ell \in s(j)} x'_{i,\ell} \; \right) \; + \; \sum_{i \in I \setminus G, \; j \in H} c_{i,j} \cdot x'_{i,j}. \label{ieq-bound-cluster-d-p-assignment}
%\end{align}
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-cost-cluster-d-prime-sum}]
%
Consider the first part of Lemma~\ref{lemma-cost-cluster-d-prime}. We have
%
\begin{align}
\sum_{i \in F^*_{D'}} y^*_i \;\; = \;\; \sum_{q \in \MC{C}_{D'}} y^*_{i(q)} \;\; \le \;\; \sum_{q \in \MC{C}_{D'}} \left( \; 2 y'^{(q)}_{i(q)} \; + \; 2\delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\} } y'^{(q)}_k \; \right).
\label{ieq-cost-cluster-d-prime-sum-1}
\end{align}
%
We charge the cost of each item in the RHS to the facilities in $I$.
%
By the design of the scaled-down operation for rounding each $q \in \MC{C}_{D'}$, we know that, for each $k \in (I \setminus G) \cap D'^{(q)}$, the facility value $y'^{(q)}_k$ decreases exactly by $\delta_{i(q)} \cdot y'^{(q)}_k$ if $k \neq i(q)$ and decreases by $y'^{(q)}_k$ followed by taken from consideration otherwise.
%
For each $k \in G \cap D'^{(q)}$, the facility value $y'^{(q)}_k$ decreases exactly by $\delta_{i(q)} \cdot y'^{(q)}_k$.
%
Hence, by charging the cost of each item in the RHS of~(\ref{ieq-cost-cluster-d-prime-sum-1}) to the value decreased for each facility, we obtain
$$\sum_{i \in F^*_{D'}} y^*_i \;\; \le \;\; 2\cdot \sum_{i \in I \setminus G} y'_i \; + \; 2\cdot \sum_{i \in G} \left( \; y'_i - y'^{(\text{II})}_i \hspace{1pt} \right).$$
%
Note that, for each $i \in G$, its facility value $y^{(\text{II})}_i$ is preserved for the second phase when it is included into a cluster in $\MC{C}_{H'}$, and thereby receives no charge from the RHS of~(\ref{ieq-cost-cluster-d-prime-sum-1}) by the charging scheme.
%

\medskip

%
The second part of this lemma follows from an analogous charging argument.
%
Consider the second part of Lemma~\ref{lemma-cost-cluster-d-prime}. We have
\begin{align}
& \sum_{i \in F^*_{D'}, \; j \in \MC{D}} c_{i,j} \cdot x^\circ_{i,j} \;\; = \;\; \sum_{q \in \MC{C}_{D'}} \; \sum_{j \in \MC{D}} \; c_{i(q),j} \cdot x^\circ_{i(q),j} \;  \notag \\[4pt]
& \hspace{1.6cm} \le \; \sum_{q \in \MC{C}_{D'}} \left( \; \sum_{j \in D} 2\cdot c_{i(q),j} \cdot x'^{(q)}_{i(q),j} \; + \; \sum_{j \in H} c_{i(q),p(j)} \cdot x'^{(q)}_{i(q),j} \; \right)  \notag \\[6pt]
& \hspace{2.2cm} + \; \sum_{q \in \MC{C}_{D'}} \delta_{i(q)} \cdot \sum_{k \in B(q) \setminus \{i(q)\}} \; \left( \; \sum_{\; j \in \MC{D}} \; 2\cdot c_{k,j} \cdot x'^{(q)}_{k,j} \; + \; \sum_{\; j \in H} c_{k,p(j)} \cdot x'^{(q)}_{k,j} \; \right)  \notag \\[6pt]
& \hspace{2.2cm} + \; \sum_{q \in \MC{C}_{D'}} \left( \;  \sum_{j \in \MC{D}} \; 2 \cdot t'_j \cdot \alpha_j \cdot x^*_{i(q),j} \; + \; \sum_{j \in H} \; 2 \cdot \alpha_j \cdot x^*_{i(q),j} \; \right).
\label{ieq-cost-cluster-d-prime-sum-2}
\end{align}
%
For the last item in~(\ref{ieq-cost-cluster-d-prime-sum-2}), we have
\begin{align*}
& \sum_{q \in \MC{C}_{D'}} \left( \;  \sum_{j \in \MC{D}} \; 2 \cdot t'_j \cdot \alpha_j \cdot x^*_{i(q),j} \; + \; \sum_{j \in H} \; 2 \cdot \alpha_j \cdot x^*_{i(q),j} \; \right) \\[8pt]
& \hspace{3cm} = \; \sum_{i \in F^*_{D'}, \; j \in \MC{D}} 2\cdot t'_j \cdot \alpha_j \cdot x^*_{i,j} \; + \; \sum_{i \in F^*_{D'}, \; j \in H} 2\cdot \alpha_j \cdot x^*_{i,j}
\end{align*}
by definition.
%
For the remaining items in the RHS of~(\ref{ieq-cost-cluster-d-prime-sum-2}), we charge the cost to the assignment values decreased due to the scaled-down operation when rounding each $q \in \MC{C}_{D'}$.
%
For each $i \in I, \; j \in \MC{D}$, the assignment value lost due to the creation of outlier clients, if present, can be partly charged except for those that become the demand of outlier clients, which is exactly $\sum_{\ell \in H(j)} x'_{i,\ell}$.
%
For $i \in G, \; j \in \MC{C} \cup H$, the assignment value that is included into a cluster in $\MC{C}_{H'}$ and preserved for the second phase, does not receive charges from the RHS of~(\ref{ieq-cost-cluster-d-prime-sum-2}).
%
Therefore, the second part of this lemma follows.
%
\end{proof}

%

\smallskip

%

%%%%
%%%%

\paragraph{The Overall Guarantee.}
\label{par-proof-approx-cfl-cfc-overall-cost}

%
In the following we establish the overall guarantee.
%
Combining Inequality~(\ref{ieq-bound-cluster-h}), Inequality~(\ref{ieq-bound-cluster-d-p-facility}), and Inequality~(\ref{ieq-bound-cluster-d-p-assignment}), 
%and applying the fact that \mong{**** changed **** no longer equal ***} $x'^{(\text{II})}_{i,j} = x^*_{i,j}$ for all $i \in G$, $j \in \MC{D} \cup H$, 
we obtain 
%
\begin{align}
\psi( \bm{x}^\circ, \bm{y}^* ) \; 
& = \; \psi \left( \left.\bm{x}^\circ\right|_{U,\MC{D}}, \left.\bm{y}^*\right|_{U} \right) \; + \; \psi \left( \left.\bm{x}^\circ\right|_{F^*_{D'},\MC{D}}, \left.\bm{y}^*\right|_{F^*_{D'}} \right) \; + \; \psi \left( \left.\bm{x}^\circ\right|_{G,\MC{D}}, \left.\bm{y}^*\right|_{G} \right)  \notag \\[8pt]
& \le \hspace{4pt} 4\cdot \sum_{i \in U} y'_i \; + \; 2\cdot \sum_{i \in I} y'_i \; + \; \sum_{i \in U, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j}  \notag \\
& \hspace{20pt} \; + \; \sum_{i \in I, \; j \in \MC{D}} 2\cdot c_{i,j} \cdot \left( \; x'_{i,j} - \sum_{\ell \in H(j)} x'_{i,\ell} \; \right) \; + \; \sum_{i \in I, \; j \in H} c_{i,p(j)} \cdot x'_{i,j}  \notag \\[8pt]
& \hspace{20pt} \; + \; \left( \;\; \sum_{j \in \MC{D}} \;\; 2 \cdot t'_j \cdot \alpha_j + \sum_{j \in H} \;\; 2\cdot \alpha_j \; \right) \cdot \left( \; \sum_{i \in F^*_{D'}} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} \; \right) .
\label{ieq-bound-overall-1}
\end{align}
%
%\begin{align}
%\Psi( \bm{x}^\circ, \bm{y}^* ) \; 
%& = \; \Psi \left( \left.\bm{x}^\circ\right|_{U,\MC{D}}, \left.\bm{y}^*\right|_{U} \right) \; + \; \Psi \left( \left.\bm{x}^\circ\right|_{F^*_{D'},\MC{D}}, \left.\bm{y}^*\right|_{F^*_{D'}} \right) \; + \; \Psi \left( \left.\bm{x}^\circ\right|_{G,\MC{D}}, \left.\bm{y}^*\right|_{G} \right)  \notag \\[8pt]
%& \le \hspace{4pt} 4\cdot \sum_{i \in U} y'_i \; + \; 2\cdot \sum_{i \in I} y'_i \; + \; \sum_{i \in U, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j}  \notag \\
%& \hspace{60pt} \; + \; \sum_{i \in G, \; j \in H} c_{i,j} \cdot x'_{i,j} \; + \; \sum_{i \in G, \; j \in \MC{D}} 2 \cdot c_{i,j} \cdot \left( \; x'_{i,j} - \sum_{\ell \in s(j)} x'_{i,\ell} \; \right)  \notag \\
%& \hspace{60pt} \; + \; \sum_{i \in I \setminus G, \; j \in \MC{D}} 2\cdot c_{i,j} \cdot \left( \; x'_{i,j} - \sum_{\ell \in s(j)} x'_{i,\ell} \; \right) \; + \; \sum_{i \in I \setminus G, \; j \in H} c_{i,j} \cdot x'_{i,j}  \notag \\[8pt]
%& \hspace{60pt} \; + \; \sum_{i \in F^*_{D'} \cup G, \; j \in \MC{D}} 2 \cdot t'_j \cdot \alpha_j \cdot x^*_{i,j} \; + \; \sum_{i \in F^*_{D'} \cup G, \; j \in H} 2 \cdot \alpha_j \cdot x^*_{i,j} .
%\label{ieq-bound-overall-1}
%\end{align}
%
%
%Consider the term $\sum_{i \in I, \; j \in H} c_{i,j} \cdot x'_{i,j}$ in the above.
%~\ref{ieq-bound-overall-1}.
% any $i \in I$ and any $j \in H$.
%
Consider the item $\sum_{i \in I, \; j \in H} c_{i,p(j)} \cdot x'_{i,j}$ in~(\ref{ieq-bound-overall-1}).
%
%By the algorithm setting, for any $j \in H$ and $i \in I$, we have $c_{i,j} \le c_{i,p(j)} + c_{w(j),p(j)}$, and
%
%it follows that \mong{**********}
By the definition of $H(j)$ for each $j \in \MC{D}$, we have
\begin{align}
\sum_{i \in I, \; j \in H} c_{i,p(j)} \cdot x'_{i,j} \; = \; \sum_{i \in I, \; j \in \MC{D}, \; \ell \in H(j)} c_{i,j} \cdot x'_{i,\ell}.
%& \le \; \sum_{i \in I, \; j \in H} c_{i,p(j)} \cdot x'_{i,j} \; + \; \sum_{i \in I, \; j \in H} c_{w(j),p(j)} \cdot x'_{i,j}  \notag \\[4pt]
%& \le \; \sum_{i \in I, \; j \in \MC{D}} \; \sum_{\ell \in s(j)} c_{i,j} \cdot x'_{i,\ell} \; + \; \sum_{i \in U, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j},
\label{ieq-bound-overall-2-1}
\end{align}
%
%where in the last inequality we use the fact that for any $j \in H$, we have
%$$\sum_{i \in I} x'_{i,j} \; = \; d_j \; := \; \frac{r'_{p(j)}}{\sum_{i \in U}x'_{i,p(j)}} \cdot x'_{w(j),p(j)} \; \le \; x'_{w(j),p(j)}.$$
% for any $j \in \MC{D}$ and any $\ell \in s(j)$,
%

\smallskip

%
\noindent
%\mong{**** polish below *****}
%Consider 
%$\sum_{i \in F^*_{D'}\cup G, \; j \in \MC{D}}2\cdot t'_j \cdot \alpha_j \cdot x^*_{i,j} + \sum_{i \in F^*_{D'} \cup G, \; j \in H} 2\cdot \alpha_j \cdot x^*_{i,j}$
%the last term in~(\ref{ieq-bound-overall-1}).
%
Consider the last item in~(\ref{ieq-bound-overall-1}).
%
Applying the definition of $t'_j$ for each $j \in \MC{D}$ with $\sum_{i \in F^*_{D'}} x^*_{i,j} + \sum_{i \in G} x'^{(\text{II})}_{i,j} > 0$, we have 
%
\begin{align}
\sum_{j \in \MC{D}} \; 2 \cdot t'_j \cdot \alpha_j \cdot \left( \; \sum_{i \in F^*_{D'}} x^*_{i,j} + \sum_{i \in G} x^{(\text{II})}_{i,j} \; \right) \; = \; \sum_{j \in \MC{D}} \; 2 \cdot \alpha_j \cdot \left( \; 1- \sum_{i \in U} x'_{i,j} - r'_j \; \right).
\label{ieq-bound-overall-2-1-1}
\end{align}
%

\smallskip

%
By the algorithm design, we have $\alpha_j = \alpha_{p(j)} + c_{w(j),p(j)}$ for any $j \in H$.
%
Further applying the fact that the demand $d_j$ of any outlier client $j \in H$ is fully-assigned when created and remains fully-assigned during the rounding process,
% after it is created,
%to facilities in $F^*_{D'} \cup G$,
% definition of $t'_j$ for each $j \in \MC{D}$, 
%we have \mong{****}
it follows that
%
\begin{align}
\sum_{j \in H} \; 2 \cdot \alpha_j \cdot \left( \; \sum_{i \in F^*_{D'}} x^*_{i,j} + \sum_{i \in G} x^{(\text{II})}_{i,j} \; \right) \; 
= & \;\; \sum_{j \in H} \; 2\cdot \left( \; \alpha_{p(j)} + c_{w(j),p(j)} \; \right) \cdot d_j  \notag \\[4pt]
\le & \;\; \sum_{j \in \MC{D}} \; 2\cdot \alpha_j \cdot r'_j \; + \; \sum_{j \in H} \; 2\cdot c_{w(j),p(j)} \cdot x'_{w(j),p(j)}  \notag \\[6pt]
\le & \;\; \sum_{j \in \MC{D}} \; 2\cdot \alpha_j \cdot r'_j \; + \; 2\cdot \sum_{i \in U, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j},
\label{ieq-bound-overall-2-1-2}
\end{align}
where in the second last inequality follows from the fact that 
$$\sum_{k \in H(j)} \alpha_{p(j)} \cdot d_k \;\; = \;\; \alpha_j \cdot \sum_{k \in H(j)} d_k \;\; = \;\; \alpha_j \cdot \sum_{k \in H(j)} r'_j \cdot \frac{x'_{w(j),p(j)}}{\sum_{i \in U} x'_{i,p(j)}} \;\; = \;\; \alpha_j \cdot r'_j$$ for all $j \in \MC{D}$ and the fact that $d_j \le x'_{w(j),p(j)}$ for any $j \in H$ by the definition of $d_j$.
% we apply the fact thatthe demand $d_j$ of any outlier client $j\in H$ is fully-assigned in the algorithm and in the last inequality we apply the fact that
%$$r'_j \; = \; \sum_{k \in s(j)} d_k \; \le \; \sum_{w \in U} x'_{w,j}$$
%

\medskip

%
\noindent
Combining~(\ref{ieq-bound-overall-2-1-1}) and~(\ref{ieq-bound-overall-2-1-2}), we have
%
\begin{align}
& \left( \;\; \sum_{j \in \MC{D}} \;\; 2 \cdot t'_j \cdot \alpha_j + \sum_{j \in H} \;\; 2\cdot \alpha_j \; \right) \cdot \left( \; \sum_{i \in F^*_{D'}} x^*_{i,j} + \sum_{i \in G} x^{(\text{II})}_{i,j} \; \right)  \notag \\[6pt]
& \hspace{60pt} \le \;\; \sum_{j \in \MC{D}} \; 2 \cdot \left( \; 1- \sum_{i \in U} x'_{i,j} \; \right) \cdot \alpha_j \; + \; 2\cdot \sum_{i \in U, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j}.
\label{ieq-bound-overall-2-2}
\end{align}
%
%where in the last inequality we apply the fact that the demand $r'_j$ of any outlier client $j\in H$ is fully-assigned in the algorithm.
%

\smallskip

%
\noindent
Further combining~(\ref{ieq-bound-overall-2-1}) and~(\ref{ieq-bound-overall-2-2}) with~(\ref{ieq-bound-overall-1}), we obtain
\begin{align}
\psi( \bm{x}^\circ, \bm{y}^* ) \; 
& \le \hspace{4pt} 4\cdot \sum_{i \in U} y'_i \; + \; 3 \cdot \sum_{i \in U, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j}  \notag \\
& \hspace{20pt} \; + \; 2\cdot \sum_{i \in I} y'_i \; + \; 2\cdot \sum_{i \in I, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j} \; + \; \sum_{j \in \MC{D}} \; 2 \cdot \left( \; 1- \sum_{i \in U} x'_{i,j} \; \right) \cdot \alpha_j .
\label{ieq-bound-overall}
\end{align}
%

\noindent
The following lemma follows from complementary slackness between 
%the primal-duality between 
$(\bm{x}',\bm{y}')$ and $(\bm{\alpha},\bm{\beta},\bm{\Gamma},\bm{\eta})$, and the fact that $0 < y'_i < 1$ for all $i \in I$.
%, Theorem~\ref{theorem-cfl-ufc-approx} is proved.

\begin{lemma} \label{lemma-bound-primal-dual-for-I}
%
$$\sum_{j \in \MC{D}} \left( \; 1 - \sum_{i \in U} x'_{i,j} \; \right) \cdot \alpha_j \;\; \le \;\; \sum_{i \in I} \; y'_i \; + \; \sum_{i \in I, \; j \in \MC{D}} \; c_{i,j} \cdot x'_{i,j}.$$
%\psi \left( \left.\bm{x}'\right|_{I,\MC{D}} , \left.\bm{y}'\right|_{I} \right) . $$
%
\end{lemma}

\begin{proof}[Proof of Lemma~\ref{lemma-bound-primal-dual-for-I}]
%
Consider any $i \in I$ and the cost incurred.
%
From the fact that $(\bm{x}',\bm{y}')$ and $(\bm{\alpha},\bm{\beta},\bm{\Gamma},\bm{\eta})$ are optimal primal and dual solutions for~\ref{LP-natural-CFL} and~\ref{LP-natural-dual}, by complementary slackness conditions, we have
$$y'_i \; + \; \sum_{j\in \MC{D}} \; c_{i,j} \cdot x'_{i,j} \;\; = \;\; y'_i \cdot \left( \; u_i \cdot \beta_i \; + \; \sum_{j \in \MC{D}} \Gamma_{i,j} \; \right) \; + \; \sum_{j \in \MC{D}} c_{i,j} \cdot x'_{i,j}$$
since $y'_i > 0$ implies that constraint~\ref{cons-D-2} is tight, and $y'_i < 1$ implies that the dual variable $\eta_i$ must be zero.
%
By the fact that $\beta_i > 0$ implies that constraint~\ref{cons-P-2} is tight and $\Gamma_{i,j} > 0$ implies that constraint~\ref{cons-P-3} is tight, the above becomes 
$$\beta_i \cdot \sum_{j\in \MC{D}} \; x'_{i,j} \; + \; \sum_{j \in \MC{D}} \; \Gamma_{i,j}\cdot x'_{i,j} \; + \; \sum_{j \in \MC{D}} \; c_{i,j} \cdot x'_{i,j}.$$
%
Finally, applying the fact that $x'_{i,j} > 0$ implies that constraint~\ref{cons-D-1} is tight, we obtain
$$y'_i \; + \; \sum_{j\in \MC{D}} \; c_{i,j} \cdot x'_{i,j} \;\; = \;\; \sum_{j \in \MC{D}} \; \alpha_j \cdot x'_{i,j}.$$
%and this lemma is proved.
%

\smallskip

%
Taking summation over all facilities in $I$, we obtain
$$\sum_{i \in I} \; y'_i \; + \; \sum_{i \in I, \; j \in \MC{D}} c_{i,j}\cdot x'_{i,j} \;\; = \;\; \sum_{j \in \MC{D}, \; i \in I} \; \alpha_j \cdot x'_{i,j} \;\; \ge \;\; \sum_{j \in \MC{D}} \left( \; 1 - \sum_{i \in U} x'_{i,j} \; \right) \cdot \alpha_j.$$
%
where in the last equality we apply constraint~\ref{cons-P-1} 
%for the fact that $\sum_{i \in \MC{F}} x'_{i,j} \ge 1$ for all
for each $j \in \MC{D}$.
%
\end{proof}

%
Applying Lemma~\ref{lemma-bound-primal-dual-for-I} on Inequality~(\ref{ieq-bound-overall}), we have
%
\begin{align*}
\psi( \bm{x}^\circ, \bm{y}^* ) \; 
%& \le \hspace{4pt} 4\cdot \sum_{i \in U} y'_i \; + \; 3 \cdot \sum_{i \in U, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j} \; + \; 4\cdot \sum_{i \in I} y'_i \; + \; 4\cdot \sum_{i \in I, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j} \\[8pt]
& \; \le \; 4 \cdot \sum_{i \in \MC{F}} y'_i \; + \; 4 \cdot \sum_{i \in \MC{F}, \; j \in \MC{D}} c_{i,j} \cdot x'_{i,j}, 
\end{align*}
%
and Theorem~\ref{theorem-cfl-ufc-approx} is proved.

%

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\section*{Acknowledgement}

The author thanks the anonymous reviewers for helpful comments on the presentation of this work and Kai-Min Chung for inspiring discussions in early stages of this work.

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