Greedy approximation for the source location problem with vertex-connectivity requirements in undirected graphs

• Authors: Toshimasa Ishii
• Venue: Journal of Discrete Algorithms
• Year: 2009

Let G=(V,E) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v@?V has a demand d(v)@?Z"+, and a cost c(v)@?R"+, where Z"+ and R"+ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices minimizing @?"v"@?"Sc(v) such that there are at least d(v) pairwise vertex-disjoint paths from S to v for each vertex v@?V-S. It is known that the problem is not approximable within a ratio of O(ln@?"v"@?"Vd(v)), unless NP has an O(N^l^o^g^l^o^g^N)-time deterministic algorithm. Also, it is known that even if every vertex has a uniform cost and d^*=4 holds, then the problem is NP-hard, where d^*=max{d(v)|v@?V}. In this paper, we consider the problem in the case where every vertex has uniform cost. We propose a simple greedy algorithm for providing a max{d^*,2d^*-6}-approximate solution to the problem in O(min{d^*,|V|}d^*|V|^2) time, while we also show that there exists an instance for which it provides no better than a (d^*-1)-approximate solution. Especially, in the case of d^*=