Journal of Statistical Physics

, Volume 45, Issue 1–2, pp 1–25 | Cite as

On the statistical mechanics of the traveling salesman problem

  • G. Baskaran
  • Yaotian Fu
  • P. W. Anderson
Articles

Abstract

We consider the statistical mechanics of the traveling salesman problem (TSP) and develop some representations to study it. In one representation the mean field theory has a simple form and brings out some of the essential features of the problem. It shows that the system has spontaneous symmetry breaking at any nonzero temperature. In general the phase progressively changes as one decreases the temperature. At low temperatures the mean field theory solution is very sensitive to any small perturbations, due to the divergence of some local susceptibilities. This critical region extends down to zero temperature. We perform the quenched average for a nonmetric TSP in the second representation and the resulting problem is more complicated than the infinite-range spin-glass problem, suggesting that the free energy landscape may be more complex. The role played by “frustration” in this problem appears explicitly through the localization property of a random matrix, which resembles the tight binding matrix of an electron in a random lattice.

Key words

spin glass N-P complete optimization problems 

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Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • G. Baskaran
    • 1
  • Yaotian Fu
    • 2
  • P. W. Anderson
    • 1
  1. 1.Department of PhysicsPrinceton UniversityPrinceton
  2. 2.Department of PhysicsUniversity of Illinois at Urbana-ChampaignUrbana

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