Complexity of the Maximum k-Path Vertex Cover Problem

Abstract

This paper introduces the maximum version of the k-path vertex cover problem, called the Maximum k-Path Vertex Cover problem ($\mathsf{{Max}}{P_k}\mathsf{VC}$ for short): A path consisting of k vertices, i.e., a path of length $k-1$ is called a k-path. If a k-path $P_k$ includes a vertex v in a vertex set S, then we say that S or v covers $P_k$. Given a graph $G = (V, E)$ and an integer s, the goal of $\mathsf{{Max}}{P_k}\mathsf{VC}$ is to find a vertex subset $S\subseteq V$ of at most s vertices such that the number of k-paths covered by S is maximized. $\mathsf{{Max}}{P_k}\mathsf{VC}$ is generally NP-hard. In this paper we consider the tractability/intractability of $\mathsf{{Max}}{P_k}\mathsf{VC}$ on subclasses of graphs: We prove that $\mathsf{{Max}}{P_3}\mathsf{VC}$ and $\mathsf{{Max}}{P_4}\mathsf{VC}$ remain NP-hard even for split graphs and for chordal graphs, respectively. Furthermore, if the input graph is restricted to graphs with constant bounded treewidth, then $\mathsf{{Max}}{P_3}\mathsf{VC}$ can be solved in polynomial time.

This work is partially supported by JSPS KAKENHI Grant Numbers JP16K16006, JP17H06287, JP17K00016, JP24106004, JP26330009, and JST CREST JPMJCR1402.