Edge covering with budget constrains

We study two related problems: the Maximum weightm′-edge cover (MWEC ) problem and the Fixed cost minimum edge cover (FCEC ) problem. In the MWEC problem, we are given an undirected simple graph G = (V,E) with integral vertex weights. The goal is to select a set U ⊆ V of maximum weight so that the number of edges with at least one endpoint in U is at most m′. Goldschmidt and Hochbaum [7] show that the problem is NP-hard and they give a 3-approximation algorithm for the problem. We present an approximation algorithm that achieves a guarantee of 2, thereby improving the bound of 3 [7]. In the FCEC problem, we are given a vertex weighted graph, a bound k, and our goal is to find a subset of vertices U of total weight at least k such that the number of edges with at least one edges in in U is minimized. A 2(1 + )-approximation for the problem follows from the work of Carnes and Shmoys [4]. We improve the approximation ratio by giving a 2-approximation algorithm for the problem. Can we get better results using methods based on linear programming? We take a first step and show that the natural LP for FCEC has an ∗Department of Computer Science, Rutgers University, Camden, NJ 08102. Partially supported by NSF grant number 1218620 . E-mail: [email protected] †Department of Computer Science, Rutgers University, Camden, NJ 08102. Partially supported by NSF grant number 1218620. E-mail: [email protected].