Maximum-Cover Source-Location Problems

  • Authors:

  • Kenya Sugihara;Hiro Ito

  • Affiliations:

  • The authors are with the Graduate School of Infomatics, Kyoto University, Kyoto-shi, 606-8501 Japan. E-mail: sugihara@lab2.kuis.kyoto-u.ac.jp, E-mail: itohiro@lab2.kuis.kyoto-u.ac.jp;The authors are with the Graduate School of Infomatics, Kyoto University, Kyoto-shi, 606-8501 Japan. E-mail: sugihara@lab2.kuis.kyoto-u.ac.jp, E-mail: itohiro@lab2.kuis.kyoto-u.ac.jp

  • Venue:
  • IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
  • Year:
  • 2006

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Abstract

Given a graph G = (V,E), a set of vertices S ⊆ V covers ν ∈ V if the edge connectivity between S and ν is at least a given number k. Vertices in S are called sources. The source location problem is a problem of finding a minimum-size source set covering all vertices of a given graph. This paper presents a new variation of the problem, called maximum-cover source-location problem, which finds a source set S with a given size p, maximizing the sum of the weight of vertices covered by S. It presents an O(np + m + n log n)-time algorithm for k = 2, where n = |V| and m = |E|. Especially it runs linear time if G is connected. This algorithm uses a subroutine for finding a subtree with the maximum weight among p-leaf trees of a given vertex-weighted tree. For the problem we give a greedy-based linear-time algorithm, which is an extension of the linear-time algorithm for finding a longest path of a given tree presented by E. W. Dijkstra around 1960. Moreover, we show some polynomial solvable cases, e.g., a given graph is a tree or (k - 1)-edge-connected, and NP-hard cases, e.g., a vertex-cost function is given or G is a digraph.