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Worst-Case and Probabilistic Analysis of Algorithms for a Location Problem

Author

Listed:
  • Gerard Cornuejols

    (Carnegie-Mellon University, Pittsburgh, Pennsylvania)

  • George L. Nemhauser

    (Cornell University, Ithaca, New York)

  • Laurence A. Wolsey

    (University of Louvain, Louvain-La-Neuve, Belgium)

Abstract

We consider a location problem whose mathematical formulation is max s { z ( S ): S ⊆ N , | S | = K }, where z ( S ) = ∑ i ∈ I max j ∈ s c ij and C = ( c ij ) is any non-negative m × n matrix with row index set I and column index set N . We show that any procedure which uses matrix C only to calculate values of the function z ( S ) cannot, with a number of values polynomial in n , guarantee to find an optimal solution. However when C is the edge-vertex incidence matrix of a graph, we show that if n is suitably large and K is fixed or does not grow too rapidly with n , the K vertices of largest degree nearly always constitute an optimal solution.

Suggested Citation

  • Gerard Cornuejols & George L. Nemhauser & Laurence A. Wolsey, 1980. "Worst-Case and Probabilistic Analysis of Algorithms for a Location Problem," Operations Research, INFORMS, vol. 28(4), pages 847-858, August.
    • Handle: RePEc:inm:oropre:v:28:y:1980:i:4:p:847-858
    DOI: 10.1287/opre.28.4.847
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    Cited by:

    1. Reuven Cohen & Mira Gonen, 2019. "On interval and circular-arc covering problems," Annals of Operations Research, Springer, vol. 275(2), pages 281-295, April.
    2. Reuven Cohen & Mira Gonen & Asaf Levin & Shmuel Onn, 2017. "On nonlinear multi-covering problems," Journal of Combinatorial Optimization, Springer, vol. 33(2), pages 645-659, February.
    3. Refael Hassin & Eitan Zemel, 1984. "On Shortest Paths in Graphs with Random Weights," Discussion Papers 593, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    4. Han, Qiaoming & Ye, Yinyu & Zhang, Hantao & Zhang, Jiawei, 2002. "On approximation of max-vertex-cover," European Journal of Operational Research, Elsevier, vol. 143(2), pages 342-355, December.
    5. A.A. Ageev & M.I. Sviridenko, 2004. "Pipage Rounding: A New Method of Constructing Algorithms with Proven Performance Guarantee," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 307-328, September.

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