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A generalization of the weighted set covering problem

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  • Jian Yang
  • Joseph Y‐T. Leung

Abstract

We study a generalization of the weighted set covering problem where every element needs to be covered multiple times. When no set contains more than two elements, we can solve the problem in polynomial time by solving a corresponding weighted perfect b‐matching problem. In general, we may use a polynomial‐time greedy heuristic similar to the one for the classical weighted set covering problem studied by D.S. Johnson [Approximation algorithms for combinatorial problems, J Comput Syst Sci 9 (1974), 256–278], L. Lovasz [On the ratio of optimal integral and fractional covers, Discrete Math 13 (1975), 383–390], and V. Chvatal [A greedy heuristic for the set‐covering problem, Math Oper Res 4(3) (1979), 233–235] to get an approximate solution for the problem. We find a worst‐case bound for the heuristic similar to that for the classical problem. In addition, we introduce a general type of probability distribution for the population of the problem instances and prove that the greedy heuristic is asymptotically optimal for instances drawn from such a distribution. We also conduct computational studies to compare solutions resulting from running the heuristic and from running the commercial integer programming solver CPLEX on problem instances drawn from a more specific type of distribution. The results clearly exemplify benefits of using the greedy heuristic when problem instances are large. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2005

Suggested Citation

  • Jian Yang & Joseph Y‐T. Leung, 2005. "A generalization of the weighted set covering problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(2), pages 142-149, March.
    • Handle: RePEc:wly:navres:v:52:y:2005:i:2:p:142-149
    DOI: 10.1002/nav.10093
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    1. V. Chvatal, 1979. "A Greedy Heuristic for the Set-Covering Problem," Mathematics of Operations Research, INFORMS, vol. 4(3), pages 233-235, August.
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    Cited by:

    1. Hao Xia & Milind Dawande & Vijay Mookerjee, 2014. "Role Refinement in Access Control: Model and Analysis," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 866-884, November.

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