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 $\alpha < 1/3$.

翻译：我们研究\textsc{$\alpha$-固定基数图划分（$\alpha$-FCGP）}问题，这是Bonnet等人[Algorithmica 2015]引入的通用局部图划分问题。该问题中，给定图$G$、两个数$k$和$p$以及参数$0\leq\alpha\leq 1$，需判断是否存在大小为$k$的顶点集$S\subseteq V$，使其覆盖函数$cov_{\alpha}(S)$至少为$p$（或最小化版本中至多为$p$）。覆盖函数$cov_{\alpha}(\cdot)$以权重$\alpha$计数恰有一个端点在$S$中的边，并以权重$1-\alpha$计数两个端点均在$S$中的边。$\alpha$-FCGP泛化了\textsc{最密$k$-子图}、\textsc{最大$k$-顶点覆盖}和\textsc{最大$(k,n-k)$-割}等基础图问题。研究$\alpha$-FCGP时一个自然的问题是：已知针对其特例（如\textsc{最大$k$-顶点覆盖}）的算法结果能否推广到更一般的情形。针对\textsc{最大$k$-顶点覆盖}问题，一种简单而有效的参数化近似[Manurangsi, SOSA 2019]与亚指数算法[Fomin et al. IPL 2011]基于贪婪顶点度排序。本文的核心见解是：贪婪顶点度排序的思想可用于设计$\alpha > 0$时的固定参数近似方案（FPT-AS），以及针对$\alpha > 1/3$的最大化问题和$\alpha < 1/3$的最小化问题在顶点-极小自由图上的亚指数时间算法。