Locating sources to meet flow demands in undirected networks

This paper deals with the problem of finding a minimum-cost vertex subset S in an undirected network such that for each vertex v we can send d(v) units of flow from S to v. Although this problem is NP-hard in general, H. Tamura, H. Sugawara, M. Sengoku, and S. Shinoda (IEICE Trans. Fund. J81-A (1998), 863-869) have presented a greedy algorithm for solving the special case with uniform costs on the vertices. We give a simpler proof on the validity of the greedy algorithm using linear programming duality and improve the running time bound from O(n(2)M(n, m)) to O(nM(n, m)), where n is the number of vertices in the network and M(n, m) denotes the time for max-flow computation in a network with n vertices and m edges. We also present an O(n(m + n log n)) time algorithm for the special case with uniform demands and arbitrary costs. (C) 2002 Elsevier Science.