Path Hitting in Acyclic Graphs

Abstract

An instance of the path hitting problem consists of two families of paths, ${\cal D}$ and ${\cal H}$, in a common undirected graph, where each path in ${\cal H}$ is associated with a non-negative cost. We refer to ${\cal D}$ and ${\cal H}$ as the sets of demand and hitting paths, respectively. When $p \in {\cal H}$ and $q \in {\cal D}$ share at least one mutual edge, we say that phits q. The objective is to find a minimum cost subset of ${\cal H}$ whose members collectively hit those of ${\cal D}$.

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.

Due to space limitations, some technical details and proofs are omitted from this extended abstract. We refer the reader to the full version of this paper (currently available online at http://www.math.tau.ac.il/~segevd), in which all missing details are provided.