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The Stochastic Traveling Salesman Problem: Finite Size Scaling and the Cavity Prediction

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Abstract

We study the random link traveling salesman problem, where lengths l ij between city i and city j are taken to be independent, identically distributed random variables. We discuss a theoretical approach, the cavity method, that has been proposed for finding the optimum tour length over this random ensemble, given the assumption of replica symmetry. Using finite size scaling and a renormalized model, we test the cavity predictions against the results of simulations, and find excellent agreement over a range of distributions. We thus provide numerical evidence that the replica symmetric solution to this problem is the correct one. Finally, we note a surprising result concerning the distribution of k th-nearest neighbor links in optimal tours, and invite a theoretical understanding of this phenomenon.

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Authors and Affiliations

  1. Los Alamos National Laboratory, CIC-3 and Center for Nonlinear Studies, MS-B258, Los Alamos, New Mexico, 87545

    Allon G. Percus

  2. Division de Physique Théorique, Institut de Physique Nucléaire, Université Paris-Sud, F-91406, Orsay Cedex, France

    Olivier C. Martin

Authors
  1. Allon G. Percus
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  2. Olivier C. Martin
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Percus, A.G., Martin, O.C. The Stochastic Traveling Salesman Problem: Finite Size Scaling and the Cavity Prediction. Journal of Statistical Physics 94, 739–758 (1999). https://doi.org/10.1023/A:1004570713967

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  • DOI: https://doi.org/10.1023/A:1004570713967

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