The multi‐integer set cover and the facility terminal cover problem

Abstract

AbstractThe facility terminal cover problem is a generalization of the vertex cover problem. The problem is to “cover” the edges of an undirected graph G = (V,E) where each edge e is associated with a non‐negative demand de. An edge e = u,v is covered if at least one of its endpoint vertices is allocated capacity of at least de. Each vertex v is associated with a non‐negative weight wv. The goal is to allocate capacity cv ≥ 0 to each vertex v so that all edges are covered and the total allocation cost, $\sum\limits_{v\in V}w_{v}c_{v}$, is minimized. A recent paper by Xu et al. [Networks 50 (2007), 118‐126], studied this problem, and presented a 2e‐ approximation algorithm for this problem for e the base of the natural logarithm. We generalize here the facility terminal cover problem to the multi‐integer set cover, and relate that problem to the set cover problem, which it generalizes, and the multi‐cover problem. We present a Δ‐approximation algorithm for the multi‐integer set cover problem, for Δ the maximum coverage. This demonstrates that even though the multi‐integer set cover problem generalizes the set cover problem, the same approximation ratio holds. In the special case of the facility terminal cover problem this yields a 2‐approximation algorithm, and with run time dominated by the sorting of the edge demands. This approximation algorithm improves considerably on the result of Xu et al. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009