Path Hitting in Acyclic Graphs

Abstract

An instance of the path hitting problem consists of two families of paths, ${\mathcal{D}}$ and ℋ, in a common undirected graph, where each path in ℋ is associated with a non-negative cost. We refer to ${\mathcal{D}}$ and ℋ as the sets of demand and hitting paths, respectively. When p∈ℋ and $q\in{\mathcal{D}}$ share at least one mutual edge, we say that p hits q. The objective is to find a minimum cost subset of ℋ whose members collectively hit those of ${\mathcal{D}}$ . In this paper we provide constant factor approximation algorithms for path hitting, confined to instances in which the underlying graph is a tree, a spider, or a star. Although such restricted settings may appear to be very simple, we demonstrate that they still capture some of the most basic covering problems in graphs. Our approach combines several novel ideas: We extend the algorithm of Garg, Vazirani and Yannakakis (Algorithmica, 18:3–20, 1997) for approximate multicuts and multicommodity flows in trees to prove new integrality properties; we present a reduction that involves multiple calls to this extended algorithm; and we introduce a polynomial-time solvable variant of the edge cover problem, which may be of independent interest.