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

Abstract

The 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 d_e. An edge e = u,v is covered if at least one of its endpoint vertices is allocated capacity of at least d_e. Each vertex v is associated with a non-negative weight w_v. The goal is to allocate capacity c_v ≥ 0 to each vertex v so that all edges are covered and the total allocation cost, , 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