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

We study the random link traveling salesman problem, where lengths lij 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.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  P. G. de Gennes,et al.  Exponents for the excluded volume problem as derived by the Wilson method , 1972 .

[4]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[5]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[6]  M. Mézard,et al.  On the statistical mechanics of optimization problems of the travelling salesman type , 1984 .

[7]  M. Mézard,et al.  Replica symmetry breaking and the nature of the spin glass phase , 1984 .

[8]  H. Orland.  Mean-field theory for optimization problems , 1985 .

[9]  S. Kirkpatrick,et al.  Configuration space analysis of travelling salesman problems , 1985 .

[10]  M. Mézard,et al.  A replica analysis of the travelling salesman problem , 1986 .

[11]  N. Sourlas.  Statistical mechanics and the travelling Salesman problem , 1986 .

[12]  Giorgio Parisi,et al.  Mean-Field Equations for the Matching and the Travelling Salesman Problems , 1986 .

[13]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[14]  W. Krauth,et al.  The Cavity Method and the Travelling-Salesman Problem , 1989 .

[15]  Giorgio Parisi,et al.  Extensive numerical simulations of weighted matchings: Total length and distribution of links in the optimal solution , 1991 .

[16]  Yves Crama,et al.  Local Search in Combinatorial Optimization , 2018, Artificial Neural Networks.

[17]  Olivier C. Martin,et al.  Combining simulated annealing with local search heuristics , 1993, Ann. Oper. Res..

[18]  Physique statistique et modeles a liens aleatoires , 1996 .

[19]  Martin,et al.  Finite size and dimensional dependence in the Euclidean traveling salesman problem. , 1996, Physical review letters.

[20]  O. Bohigas,et al.  The random link approximation for the Euclidean traveling salesman problem , 1996, cond-mat/9607080.

[21]  David S. Johnson,et al.  Asymptotic experimental analysis for the Held-Karp traveling salesman bound , 1996, SODA '96.

[22]  Voyageur de commerce et problemes stochastiques associes , 1997 .

[23]  O. Martin,et al.  Comparing mean field and Euclidean matching problems , 1998, cond-mat/9803195.