A PTAS for the minimum weight connected vertex cover [formula omitted] problem on unit disk graphs

Abstract

Let G=(V,E) be a weighted graph, i.e., with a vertex weight function w:V→R+. We study the problem of determining a minimum weight connected subgraph of G that has at least one vertex in common with all paths of length two in G. It is known that this problem is NP-hard for general graphs. We first show that it remains NP-hard when restricted to unit disk graphs. Our main contribution is a polynomial time approximation scheme for this problem if we assume that the problem is c-local and the unit disk graphs have minimum degree of at least two.