Comparing mean field and Euclidean matching problems

Abstract:Combinatorial optimization is a fertile testing ground for statistical physics methods developed in the context of disordered systems, allowing one to confront theoretical mean field predictions with actual properties of finite dimensional systems. Our focus here is on minimum matching problems, because they are computationally tractable while both frustrated and disordered. We first study a mean field model taking the link lengths between points to be independent random variables. For this model we find perfect agreement with the results of a replica calculation, and give a conjecture. Then we study the case where the points to be matched are placed at random in a d-dimensional Euclidean space. Using the mean field model as an approximation to the Euclidean case, we show numerically that the mean field predictions are very accurate even at low dimension, and that the error due to the approximation is O(1/d2). Furthermore, it is possible to improve upon this approximation by including the effects of Euclidean correlations among k link lengths. Using k=3 (3-link correlations such as the triangle inequality), the resulting errors in the energy density are already less than at . However, we argue that the dimensional dependence of the Euclidean model's energy density is non-perturbative, i.e., it is beyond all orders in k of the expansion in k-link correlations.

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

[2]  Kazuoki Azuma.  WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES , 1967 .

[3]  P. R. Bevington,et al.  Data Reduction and Error Analysis for the Physical Sciences , 1969 .

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

[5]  S. Edwards,et al.  Theory of spin glasses , 1975 .

[6]  Richard M. Karp,et al.  A Patching Algorithm for the Nonsymmetric Traveling-Salesman Problem , 1979, SIAM J. Comput..

[7]  G. Parisi.  A sequence of approximated solutions to the S-K model for spin glasses , 1980 .

[8]  G. Parisi.  The order parameter for spin glasses: a function on the interval 0-1 , 1980 .

[9]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[10]  J. Steele.  Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability , 1981 .

[11]  Ulrich Derigs,et al.  An analysis of alternative strategies for implementing matching algorithms , 1983, Networks.

[12]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

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

[14]  János Komlós,et al.  On optimal matchings , 1984, Comb..

[15]  A. Bray,et al.  Lower critical dimension of Ising spin glasses: a numerical study , 1984 .

[16]  Fisher,et al.  Scaling in spin-glasses. , 1985, Physical review letters.

[17]  A. Bray,et al.  Phase diagrams for dilute spin glasses , 1985 .

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

[19]  The 1/d expansion of the Eden model , 1985 .

[20]  M. Mézard,et al.  Replicas and optimization , 1985 .

[21]  K. Binder,et al.  Spin glasses: Experimental facts, theoretical concepts, and open questions , 1986 .

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

[23]  Replica symmetric solutions in the Ising spin glass: the tree approximation , 1987 .

[24]  M. Mézard,et al.  On the solution of the random link matching problems , 1987 .

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

[26]  J. Steele.  Probability theory and combinatorial optimization , 1987 .

[27]  Fisher,et al.  Equilibrium behavior of the spin-glass ordered phase. , 1988, Physical review. B, Condensed matter.

[28]  Giorgio Parisi,et al.  The Euclidean matching problem , 1988 .

[29]  Warren D. Smith.  Studies in computational geometry motivated by mesh generation , 1989 .

[30]  Peter Grassberger,et al.  An efficient heuristic algorithm for minimum matching , 1990, ZOR Methods Model. Oper. Res..

[31]  Georges,et al.  Low-temperature phase of the Ising spin glass on a hypercubic lattice. , 1990, Physical review letters.

[32]  I. Kondor,et al.  Short-range corrections to the order parameter of the Ising spin glass above the upper critical dimension , 1991 .

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

[34]  Michel Talagrand,et al.  Matching Random Samples in Many Dimensions , 1992 .

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

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

[38]  G. Parisi.  A Conjecture on random bipartite matching , 1998, cond-mat/9801176.

[39]  O. Martin,et al.  The Stochastic Traveling Salesman Problem: Finite Size Scaling and the Cavity Prediction , 1998, cond-mat/9802295.

[40]  Thomas de Quincey.  [C] , 2000, The Works of Thomas De Quincey, Vol. 1: Writings, 1799–1820.