In the Partial Vertex Cover (PVC) problem, we are given an $n$-vertex graph $G$ and a positive integer $k$, and the objective is to find a vertex subset $S$ of size $k$ maximizing the number of edges with at least one end-point in $S$. This problem is W[1]-hard on general graphs, but admits a parameterized subexponential time algorithm with running time $2^{O(\sqrt{k})}n^{O(1)}$ on planar and apex-minor free graphs [Fomin et al. (FSTTCS 2009, IPL 2011)], and a $k^{O(k)}n^{O(1)}$ time algorithm on bounded degeneracy graphs [Amini et al. (FSTTCS 2009, JCSS 2011)]. Graphs of bounded degeneracy contain many sparse graph classes like planar graphs, $H$-minor free graphs, and bounded tree-width graphs. In this work, we prove the following results: 1) There is an algorithm for PVC with running time $2^{O(k)}n^{O(1)}$ on graphs of bounded degeneracy which is an improvement on the previous $k^{O(k)}n^{O(1)}$ time algorithm by Amini et al. 2) PVC admits a polynomial compression on graphs of bounded degeneracy, resolving an open problem posed by Amini et al.
翻译:在部分顶点覆盖( PVC) 问题中, 我们在平面和平面最小值图上被给了 $n$- verdex 图形$G$和正整数 美元, 目标是找到一个大小为S$S$的顶端子子集$S$, 使边缘数量最大化, 至少有一个终点为$S$。 这个问题在一般图形上是 W[ 1 硬的, 但在一般图形中承认了一个参数化的亚特异化时间算法, 运行时间为 2°O (\ sqrt{k}} 硬度( O) 硬度图, 硬度图( $H$) 和绑定的树维值图 [FometCS, 2009, IPL, 2011], 和 折叠式离子子子子子子子子子子集 。 在这项工作中, A- (x) 的A- li lixal_ 上, 我们用时间算算出一个结果。