Locating sources to meet flow demands in undirected networks

  • Authors:

  • Kouji Arata;Satoru Iwata;Kazuhisa Makino;Satoru Fujishige

  • Affiliations:

  • Matsushita Communication Industrial Co., Ltd., Yokohama 223-8639, Japan;Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-0033, Japan;Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan;Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

  • Venue:
  • Journal of Algorithms
  • Year:
  • 2002

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Abstract

This paper deals with the problem of finding a minimum-cost vertex subset S in an undirected network such that for each vertex υ we can send d(υ) units of flow from S to υ. 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(n2M(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.