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

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 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, ∑ v∈V wvcv, is minimized. A recent paper by Xu, Yang and Xu (Networks, 50, (2007), 118-126), studied this problem, and presented a 2eapproximation 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 ∆-approximation algorithm for the multi-integer set cover problem, for ∆ 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, Yang and Xu.