Path hitting in acyclic graphs

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