  First, we study the Unconstrained Fault-Tolerant Resource Allocation (UFTRA)
problem (a.k.a. FTFA problem in \cite{shihongftfa}). In the problem, we are
given a set of sites equipped with an unconstrained number of facilities as
resources, and a set of clients with set $\mathcal{R}$ as corresponding
connection requirements, where every facility belonging to the same site has an
identical opening (operating) cost and every client-facility pair has a
connection cost. The objective is to allocate facilities from sites to satisfy
$\mathcal{R}$ at a minimum total cost. Next, we introduce the Constrained
Fault-Tolerant Resource Allocation (CFTRA) problem. It differs from UFTRA in
that the number of resources available at each site $i$ is limited by $R_{i}$.
Both problems are practical extensions of the classical Fault-Tolerant Facility
Location (FTFL) problem \cite{Jain00FTFL}. For instance, their solutions
provide optimal resource allocation (w.r.t. enterprises) and leasing (w.r.t.
clients) strategies for the contemporary cloud platforms.
  In this paper, we consider the metric version of the problems. For UFTRA with
uniform $\mathcal{R}$, we present a star-greedy algorithm. The algorithm
achieves the approximation ratio of 1.5186 after combining with the cost
scaling and greedy augmentation techniques similar to
\cite{Charikar051.7281.853,Mahdian021.52}, which significantly improves the
result of \cite{shihongftfa} using a phase-greedy algorithm. We also study the
capacitated extension of UFTRA and give a factor of 2.89. For CFTRA with
uniform $\mathcal{R}$, we slightly modify the algorithm to achieve
1.5186-approximation. For a more general version of CFTRA, we show that it is
reducible to FTFL using linear programming.
