  We study the \textsc{$\alpha$-Fixed Cardinality Graph Partitioning
($\alpha$-FCGP)} problem, the generic local graph partitioning problem
introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given
a graph $G$, two numbers $k,p$ and $0\leq\alpha\leq 1$, the question is whether
there is a set $S\subseteq V$ of size $k$ with a specified coverage function
$cov_{\alpha}(S)$ at least $p$ (or at most $p$ for the minimization version).
The coverage function $cov_{\alpha}(\cdot)$ counts edges with exactly one
endpoint in $S$ with weight $\alpha$ and edges with both endpoints in $S$ with
weight $1 - \alpha$. $\alpha$-FCGP generalizes a number of fundamental graph
problems such as \textsc{Densest $k$-Subgraph}, \textsc{Max $k$-Vertex Cover},
and \textsc{Max $(k,n-k)$-Cut}.
  A natural question in the study of $\alpha$-FCGP is whether the algorithmic
results known for its special cases, like \textsc{Max $k$-Vertex Cover}, could
be extended to more general settings. One of the simple but powerful methods
for obtaining parameterized approximation [Manurangsi, SOSA 2019] and
subexponential algorithms [Fomin et al. IPL 2011] for \textsc{Max $k$-Vertex
Cover} is based on the greedy vertex degree orderings. The main insight of our
work is that the idea of greed vertex degree ordering could be used to design
fixed-parameter approximation schemes (FPT-AS) for $\alpha > 0$ and the
subexponential-time algorithms for the problem on apex-minor free graphs for
maximization with $\alpha > 1/3$ and minimization with $\alpha < 1/3$.
