Advertisement

Finding the Exact Integrality Gap for Small Traveling Salesman Problems

  • Sylvia Boyd
  • Geneviève Labonté
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)

Abstract

The Symmetric Traveling Salesman Problem (STSP) is to find a minimum weight Hamiltonian cycle in a weighted complete graph on n nodes. One direction which seems promising for finding improved solutions for the STSP is the study of a linear relaxation of this problem called the Subtour Elimination Problem (SEP). A well known conjecture in combinatorial optimization says that the integrality gap of the SEP is 4/3 in the metric case. Currently the best upper bound known for this integrality gap is 3/2.

Finding the exact value for the integrality gap for the SEP is difficult even for small values of n due to the exponential size of the data involved. In this paper we describe how we were able to overcome such difficulties and obtain the exact integrality gap for all values of n up to 10. Our results give a verification of the 4/3 conjecture for small values of n, and also give rise to a new stronger form of the conjecture which is dependent on n.

Keywords

Integer Linear Programming Travel Salesman Problem Complementary Slackness Support Graph Binary Integer Programming 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Boyd, S., Carr. R. (1999): A new bound for the ratio between the 2-matching problem and its linear programming relaxation, Math. Prog. Series A 86, 499–514.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Boyd, S., Carr, R. (2000): Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices, Technical Report TR-96-12, University of Ottawa, Ottawa.Google Scholar
  3. 3.
    Boyd, S., Pulleyblank, W.R. (1991): Optimizing over the subtour polytope of the traveling salesman problem, Math. Prog. 49, 163–187.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Carr, R., Ravi, R. (1998): A new bound for the 2-edge connected subgraph problem, Proceedings of the Conference on Integer Programming and Combinatorial Optimization (IPCO’98).Google Scholar
  5. 5.
    Christof, T., Löbel, A, Stoer, M. (1997): PORTA, A POlyhedron Representation Transformation Algorithm, http://www.zib.de/Optimization/Software/Porta/index.html
  6. 6.
    Christofides, N. (1976): Worst case analysis of a new heuristic for the traveling salesman problem, Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh.Google Scholar
  7. 7.
    Grötschel, M., Lovasz, L., Schrijver, A. (1988): Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, Berlin.zbMATHCrossRefGoogle Scholar
  8. 8.
    Johnson, D.S., Papadimitriou, C.H. (1985): Computational Complexity, In: Lawler et al, eds., The Traveling Salesman Problem, John Wiley & Sons, Chichester, 37–85.Google Scholar
  9. 9.
    McKay, B. (1991): nauty User’s Guide (Version 1.5), Technical Report TR-CS-90-02, Department of Computer Science, Australia National University.Google Scholar
  10. 10.
    Shmoys, D.B., Williamson, D.P. (1990): Analyzing the Held-Karp TSP bound: A monotonicity property with application, Inf. Process. Lett. 35, 281–285.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Woolsey, L.A. (1980): Heuristic analysis, linear programming and branch and bound, Math. Prog. Study 13, 121–134.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Sylvia Boyd
    • 1
  • Geneviève Labonté
    • 1
  1. 1.University of OttawaOttawaCanada

Personalised recommendations