TSP on Cubic and Subcubic Graphs

  • Sylvia Boyd
  • René Sitters
  • Suzanne van der Ster
  • Leen Stougie
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6655)

Abstract

We study the Travelling Salesman Problem (TSP) on the metric completion of cubic and subcubic graphs, which is known to be NP-hard. The problem is of interest because of its relation to the famous 4/3 conjecture for metric TSP, which says that the integrality gap, i.e., the worst case ratio between the optimal values of the TSP and its linear programming relaxation, is 4/3. Using polyhedral techniques in an interesting way, we obtain a polynomial-time 4/3-approximation algorithm for this problem on cubic graphs, improving upon Christofides’ 3/2-approximation, and upon the 3/2 − 5/389 ≈ 1.487-approximation ratio by Gamarnik, Lewenstein and Svirdenko for the case the graphs are also 3-edge connected. We also prove that, as an upper bound, the 4/3 conjecture is true for this problem on cubic graphs. For subcubic graphs we obtain a polynomial-time 7/5-approximation algorithm and a 7/5 bound on the integrality gap.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aggarwal, N., Garg, N., Gupta, S.: A 4/3-approximation for TSP on cubic 3-edge-connected graphs (2011) (manuscript) Google Scholar
  2. 2.
    Akiyama, T., Nishizeki, T., Saito, N.: NP-completeness of the hamiltonian cycle problem for bipartite graphs. Journal of Information Processing 3, 73–76 (1980)MathSciNetMATHGoogle Scholar
  3. 3.
    Arora, S., Grigni, M., Karger, D., Klein, P., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Proc. of the 9th ACM–SIAM Symposium on Discrete Algorithms, pp. 33–41 (1998)Google Scholar
  4. 4.
    Barahona, F.: Fractional packing of T-joins. SIAM Journal on Discrete Math. 17, 661–669 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Benoit, G., Boyd, S.: Finding the exact integrality gap for small travelling salesman problems. Math. of Operations Research 33, 921–931 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Berman, P., Karpinski, M.: 8/7-approximation algorithm for 1,2-TSP. In: Proc. 17th ACM SIAM Symposium on Discrete Algorithms, pp. 641–648 (2006)Google Scholar
  7. 7.
    Boyd, S., Iwata, S., Takazawa, K.: Finding 2-Factors Covering 3- and 4-Edge Cuts in Bridgeless Cubic Graphs Kyoto University (2010) (manuscript)Google Scholar
  8. 8.
    Csaba, B., Karpinski, M., Krysta, P.: Approximability of dense and sparse instances of minimum 2-connectivity, tsp and path problems. In: Proc. 13th ACM–SIAM Symposium on Discrete Algorithms, pp. 74–83 (2002)Google Scholar
  9. 9.
    Christofides, N.: Worst case analysis of a new heuristic for the traveling salesman problem, Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh (1976)Google Scholar
  10. 10.
    Cornuéjols, G., Fonlupt, J., Naddef, D.: The traveling salesman on a graph and some related integer polyhedra. Math. Programming 33, 1–27 (1985)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. J. of Res. National Bureau of Standards B 69, 125–130 (1965)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fotakis, D., Spirakis, P.: Graph properties that facilitate travelling. Electronic Colloquium on Computational Complexity 31, 1–18 (1998)Google Scholar
  13. 13.
    Fulkerson, D.: Blocking and anti-blocking pairs of polyhedra. Math Programming 1, 168–194 (1971)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gamarnik, D., Lewenstein, M., Sviridenko, M.: An improved upper bound for the TSP in cubic 3-edge-connected graphs. OR Letters 33, 467–474 (2005)MathSciNetMATHGoogle Scholar
  15. 15.
    Garey, M., Johnson, D., Tarjan, R.: The planar hamiltonian circuit problem is NP-complete. SIAM Journal of Computing 5, 704–714 (1976)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gharan, S.O., Saberi, A., Singh, M.: A Randomized Rounding Approach to the Traveling Salesman Problem (2011) (manuscript)Google Scholar
  17. 17.
    Grigni, M., Koutsoupias, E., Papadimitriou, C.: An approximation scheme for planar graph TSP. In: Proc. 36th Annual Symposium on Foundations of Computer Science, pp. 640–645 (1995)Google Scholar
  18. 18.
    Hartvigsen, D., Li, Y.: Maximum cardinality simple 2-matchings in subcubic graphs, University of Notre Dame (2009) (manuscript)Google Scholar
  19. 19.
    Kaiser, T., Král’, D., Norine, S.: Unions of perfect matchings in cubic graphs. Electronic Notes in Discrete Math. 22, 341–345 (2005)CrossRefMATHGoogle Scholar
  20. 20.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: The Traveling Salesman Problem–A Guided Tour of Combinatorial Optimization. Wiley, Chichester (1985)MATHGoogle Scholar
  21. 21.
    Naddef, D., Pulleyblank, W.: Matchings in regular graphs. Discrete Math. 34, 283–291 (1981)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Papadimitriou, C., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res. 18, 1–11 (1993)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Petersen, J.: Die Theorie der regulären graphen. Acta Math. 15, 193–220 (1891)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shmoys, D., Williamson, D.: Analyzing the Held-Karp TSP bound: A monotonicity property with application. Information Processing Letters 35, 281–285 (1990)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wolsey, L.: Heuristic analysis, linear programming and branch and bound. Math. Programming Study 13, 121–134 (1980)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Sylvia Boyd
    • 1
  • René Sitters
    • 2
  • Suzanne van der Ster
    • 2
  • Leen Stougie
    • 2
    • 3
  1. 1.School of Information Technology and Engineering (SITE)University of OttawaOttawaCanada
  2. 2.Department of Operations ResearchVU UniversityAmsterdamThe Netherlands
  3. 3.CWIAmsterdamThe Netherlands

Personalised recommendations