Path Hitting in Acyclic Graphs

An instance of the path hitting problem consists of two families of paths, D and H, in a common undirected graph, where each path in H is associated with a non-negative cost. We refer to D and H as the sets of demand and hitting paths, respectively. When p ∈ H and q ∈ 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 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.