FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges

We study the \textsc{$α$-Fixed Cardinality Graph Partitioning ($α$-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 $0\leqα\leq 1$, the question is whether there is a set $S\subseteq V$ of size $k$ with a specified coverage function $cov_α(S)$ at least $p$ (or at most $p$ for the minimization version). The coverage function $cov_α(\cdot)$ counts edges with exactly one endpoint in $S$ with weight $α$ and edges with both endpoints in $S$ with weight $1 - α$. $α$-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 $α$-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 $α> 0$ and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with $α> 1/3$ and minimization with $α< 1/3$.