Acta Mathematica

The mean field traveling salesman and related problems

Johan Wästlund

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Abstract

The edges of a complete graph on n vertices are assigned i.i.d. random costs from a distribution for which the interval [0, t] has probability asymptotic to t as t→0 through positive values. In this so called pseudo-dimension 1 mean field model, we study several optimization problems, of which the traveling salesman is the best known. We prove that, as n→∞, the cost of the minimum traveling salesman tour converges in probability to a certain number, approximately 2.0415, which is characterized analytically.

Article information

Source
Acta Math., Volume 204, Number 1 (2010), 91-150.

Dates
Received: 3 April 2008
Revised: 9 June 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892445

Digital Object Identifier
doi:10.1007/s11511-010-0046-7

Mathematical Reviews number (MathSciNet)
MR2600434

Zentralblatt MATH identifier
1231.90370

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 90C35: Programming involving graphs or networks [See also 90C27]

Rights
2010 © Institut Mittag-Leffler

Citation

Wästlund, Johan. The mean field traveling salesman and related problems. Acta Math. 204 (2010), no. 1, 91--150. doi:10.1007/s11511-010-0046-7. https://projecteuclid.org/euclid.acta/1485892445


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References

  • A chlioptas, D., N aor, A. & P eres, Y., Rigorous location of phase transitions in hard optimization problems. Nature, 435 (2005), 759–764.
  • A chlioptas, D. & P eres, Y., The threshold for random k-SAT is 2k log 2−O(k). J. Amer. Math. Soc., 17 (2004), 947–973.
  • A ldous, D., Asymptotics in the random assignment problem. Probab. Theory Related Fields, 93 (1992), 507–534.
  • _____ The ζ(2) limit in the random assignment problem. Random Structures Algorithms, 18 (2001), 381–418.
  • _____ Percolation-like scaling exponents for minimal paths and trees in the stochastic mean field model. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 825–838.
  • A ldous, D. & B andyopadhyay, A., A survey of max-type recursive distributional equations. Ann. Appl. Probab., 15 (2005), 1047–1110.
  • A ldous, D. & P ercus, A. G., Scaling and universality in continuous length combinatorial optimization. Proc. Natl. Acad. Sci. USA, 100:20 (2003), 11211–11215.
  • A ldous, D. & S teele, J. M., The objective method: probabilistic combinatorial optimization and local weak convergence, in Probability on Discrete Structures, Encyclopaedia Math. Sci., 110, pp. 1–72. Springer, Berlin–Heidelberg, 2004.
  • A lm, S.E. & S orkin, G. B., Exact expectations and distributions for the random assignment problem. Combin. Probab. Comput., 11 (2002), 217–248.
  • B acci, S. & M iranda, E. N., The traveling salesman problem and its analogy with two-dimensional spin glasses. J. Stat. Phys., 56 (1989), 547–551.
  • B andyopadhyay, A. & G amarnik, D., Counting without sampling: asymptotics of the log-partition function for certain statistical physics models. Random Structures Algorithms, 33 (2008), 452–479.
  • B oettcher, S. & P ercus, A., Nature’s way of optimizing. Artificial Intelligence, 119 (2000), 275–286.
  • B runetti, R., K rauth, W., M ézard, M. & P arisi, G., Extensive numerical simulation of weighted matchings: total length and distribution of links in the optimal solution. Europhys. Lett., 14 (1991), 295–301.
  • B uck, M. W., C han, C. S. & R obbins, D. P., On the expected value of the minimum assignment. Random Structures Algorithms, 21 (2002), 33–58.
  • C erf, N. J., B outet de Monvel, J., B ohigas, O., M artin, O. C. & P ercus, A. G., The random link approximation for the Euclidean traveling salesman problem. J. Physique, 7 (1997), 117–136.
  • C oppersmith, D. & S orkin, G. B., Constructive bounds and exact expectations for the random assignment problem. Random Structures Algorithms, 15 (1999), 113–144.
  • F rieze, A., On the value of a random minimum spanning tree problem. Discrete Appl. Math., 10 (1985), 47–56.
  • _____ On random symmetric travelling salesman problems. Math. Oper. Res., 29 (2004), 878–890.
  • G uerra, F., Broken replica symmetry bounds in the mean field spin glass model. Comm. Math. Phys., 233 (2003), 1–12.
  • van der H ofstad, R., H ooghiemstra, G. & Van M ieghem, P., The weight of the shortest path tree. Random Structures Algorithms, 30 (2007), 359–379.
  • K arp, R. M., An upper bound on the expected cost of an optimal assignment, in Discrete Algorithms and Complexity (Kyoto, 1986), Perspect. Comput., 15, pp. 1–4. Academic Press, Boston, MA, 1987.
  • K irkpatrick, S. & T oulouse, G., Configuration space analysis of travelling salesman problems. J. Physique, 46:8 (1985), 1277–1292.
  • K rauth, W. & M ézard, M., The cavity method and the travelling salesman problem. Europhys. Lett., 8 (1989), 213–218.
  • L inusson, S. & W ästlund, J., A proof of Parisi’s conjecture on the random assignment problem. Probab. Theory Related Fields, 128 (2004), 419–440.
  • M ézard, M. & M ontanari, A., Information, Physics, and Computation. Oxford Graduate Texts. Oxford University Press, Oxford, 2009.
  • M ézard, M. & P arisi, G., Replicas and optimization. J. Physique, 46 (1985), 771–778.
  • _____ Mean-field equations for the matching and the travelling salesman problems. Europhys. Lett., 2 (1986), 913–918.
  • _____ A replica analysis of the travelling salesman problem. J. Physique, 47 (1986), 1285–1296.
  • _____ On the solution of the random link matching problems. J. Physique, 48 (1987), 1451–1459.
  • M ézard, M., P arisi, G. & V irasoro, M. A., Spin Glass Theory and Beyond. World Scientific Lecture Notes in Physics, 9. World Scientific, Teaneck, NJ, 1987.
  • M ézard, M., P arisi, G. & Z ecchina, R., Analytic and algorithmic solutions of random satisfiability problems. Science, 297 (2002), 812.
  • N air, C., Proofs of the Parisi and Coppersmith–Sorkin Conjectures in the Finite Random Assignment Problem. Ph.D. Thesis, Stanford University, Stanford, CA, 2005.
  • N air, C., P rabhakar, B. & S harma, M., Proofs of the Parisi and Coppersmith–Sorkin random assignment conjectures. Random Structures Algorithms, 27 (2005), 413–444.
  • P arisi, G., A sequence of approximated solutions to the S–K model for spin glasses. J. Physique, 13 (1980), L–115.
  • _____ Spin glasses and optimization problems without replicas, in Le hasard et la matière (Les Houches, 1986), pp. 525–552. North-Holland, Amsterdam, 1987.
  • _____ A conjecture on random bipartite matching. Preprint, 1998. arXiv:cond-mat/9801176.
  • P ercus, A. G., Voyageur de commerce et problèmes stochastiques associés. Ph.D. Thesis, Université Pierre et Marie Curie, Paris, 1997.
  • P ercus, A. G. & M artin, O. C., Finite size and dimensional dependence in the Euclidean traveling salesman problem. Phys. Rev. Lett., 76 (1996), 1188–1191.
  • _____ The stochastic traveling salesman problem: Finite size scaling and the cavity prediction. J. Stat. Phys., 94 (1999), 739–758.
  • S herrington, D. & K irkpatrick, S., Solvable model of a spin glass. Phys. Rev. Lett., 35 (1975), 1792–1796.
  • S ourlas, N., Statistical mechanics and the travelling salesman problem. Europhys. Lett., 2 (1986), 919–923.
  • T alagrand, M., Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math., 81 (1995), 73–205.
  • ____ Spin Glasses: A Challenge for Mathematicians. Modern Surveys in Mathematics, 46. Springer, Berlin–Heidelberg, 2003.
  • ____ The Parisi formula. Ann. of Math., 163 (2006), 221–263.
  • V annimenus, J. & M ézard, M., On the statistical mechanics of optimization problems of the travelling salesman type. J. Physique, 45 (1984), 1145–1153.
  • W alkup, D. W., On the expected value of a random assignment problem. SIAM J. Comput., 8 (1979), 440–442.
  • W ästlund, J., A proof of a conjecture of Buck, Chan, and Robbins on the expected value of the minimum assignment. Random Structures Algorithms, 26 (2005), 237–251.
  • _____ The variance and higher moments in the random assignment problem. Linköping Studies in Mathematics, 8. Preprint, 2005.
  • W eitz, D., Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 140–149. ACM, New York, 2006.