We study the 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 $0\le \alpha \le 1$, the question is whether there is a set $S\subseteq V$ of size k with a specified coverage function ${\mathsf{cov}}_{\alpha }(S)$ at least p (or at most p for the minimization version). The coverage function ${\mathsf{cov}}_{\alpha }(\cdot )$ counts edges with exactly one endpoint in S with weight α and edges with both endpoints in S with weight $1-\alpha$. α-FCGP generalizes a number of fundamental graph problems such as Densest k-Subgraph, Max k-Vertex Cover, and 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 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 Max k-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greedy vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for $\alpha >0$ and subexponential-time algorithms for the problem on apex-minor free graphs for maximization with $\alpha >1/3$ and minimization with $\alpha <1/3$.⁴

我们研究了α-固定基数图划分（α -FCGP）问题，这是 Bonnet 等人提出的通用局部图划分问题。[算法 2015]。在这个问题中，我们给出一个图G，两个数字$k,p$和$0\le \alpha \le 1$，问题是是否存在一个集合$S\subseteq V$大小为k且具有指定的覆盖函数${\mathsf{冠状病毒}}_{\alpha }(S）$至少p （或对于最小化版本至多p ）。覆盖函数${\mathsf{冠状病毒}}_{\alpha }(\cdot ）$计算S中正好有一个端点且权重为α 的边，以及计算两个端点都在S中且权重为 α 的边$1-\alpha$。α -FCGP 概括了许多基本图问题，例如Densest k -Subgraph、Max k -Vertex Cover和Max $(k,n-k）$-切。

α -FCGP研究中的一个自然问题是，以其特殊情况而闻名的算法结果（例如Max k -Vertex Cover）是否可以扩展到更一般的设置。获得参数化近似的简单但强大的方法之一 [Manurangsi, SOSA 2019] 和次指数算法 [Fomin 等人。IPL 2011] Max k -Vertex Cover基于贪婪顶点度排序。我们工作的主要见解是，贪婪顶点度排序的思想可用于设计固定参数近似方案（FPT-AS）：$\alpha >0$以及用于最大化小顶点自由图问题的次指数时间算法$\alpha >1/3$和最小化$\alpha <1/3$。⁴