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A vertex‐closing approach to the p‐center problem

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  • Joseph S. Martinich

Abstract

This article uses a vertex‐closing approach to investigate the p‐center problem. The optimal set of vertices to close are found in imbedded subgraphs of the original graph. Properties of these subgraphs are presented and then used to characterize the optimal solution, to establish a priori upper and lower bounds, to establish refined lower bounds, and to verify the optimality of solutions. These subgraphs form the foundation of two polynomial algorithms of complexity O(|E| log |E|) and O(|E|2). The algorithms are proven to converge to an optimum for special cases, and computational evidence is provided which suggests that they produce very good solutions more generally. Both algorithms perform very well on problems where p is large relative to the number of vertices n, specifically, when p/n ≥ 0.30. One of the algorithms is especially efficient for solving a sequence of problems on the same graph.

Suggested Citation

  • Joseph S. Martinich, 1988. "A vertex‐closing approach to the p‐center problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(2), pages 185-201, April.
    • Handle: RePEc:wly:navres:v:35:y:1988:i:2:p:185-201
    DOI: 10.1002/1520-6750(198804)35:23.0.CO;2-R
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    3. S. L. Hakimi, 1965. "Optimum Distribution of Switching Centers in a Communication Network and Some Related Graph Theoretic Problems," Operations Research, INFORMS, vol. 13(3), pages 462-475, June.
    4. A. J. Goldman, 1972. "Minimax Location of a Facility in a Network," Transportation Science, INFORMS, vol. 6(4), pages 407-418, November.
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