Shorter tours by nicer ears: 7/5-approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs

  • András SebőEmail author
  • Jens VygenEmail author


We prove new results for approximating the graph-TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees.

For the graph-TSP itself, we improve the approximation ratio to 7/5. For a generalization, the minimum T-tour problem, we obtain the first nontrivial approximation algorithm, with ratio 3/2. This contains the s-t-path graph-TSP as a special case. Our approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4/3.

The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.

Mathematics Subject Classification (2000)

90C27 05C85 68R10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Alexander, S. Boyd, and P. Elliott-Magwood: On the integrality gap of the 2-edge connected subgraph problem, Technical Report TR-2006-04, SITE, University of Ottawa, 2006.Google Scholar
  2. [2]
    H.-C. An, R. Kleinberg, and D. B. Shmoys: Improving Christofides’ algorithm for the s-t path TSP, Proceedings of the 44th Annual ACM Symposium on Theory of Computing (2012), 875–886Google Scholar
  3. [3]
    F. Barahona and M. Conforti: A construction for binary matroids, Discrete Mathematics 66 (1987), 213–218.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    S. Boyd, R. Sitters, S. van der Ster and L. Stougie: TSP on cubic and subcubic graphs, in: Integer Programming and Combinatorial Optimization; Proceedings of the 15th IPCO Conference; LNCS 6655 (O. Günlük, G.J. Woeginger, eds.), Springer Berlin 2011, 65–77.Google Scholar
  5. [5]
    R. Carr and R. Ravi: A new bound for the 2-edge connected subgraph problem, in: Integer Programming and Combinatorial Optimization; Proceedings of the 6th IPCO Conference; LNCS 1412 (R.E. Bixby, E.A. Boyd, R.Z. Ríos-Mercado, eds.), Springer Berlin 1998, 112–125.CrossRefGoogle Scholar
  6. [6]
    J. Cheriyan, A. Sebö and Z. Szigeti: Improving on the 1.5-approximation of a smallest 2-edge connected spanning subgraph, SIAM Journal on Discrete Mathematics 14 (2001), 170–180.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    N. Christofides: Worst-case analysis of a new heuristic for the traveling salesman problem, Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University Pittsburgh 1976.Google Scholar
  8. [8]
    W. J. Cook: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation, Princeton University Press 2012.Google Scholar
  9. [9]
    G. Cornuéjols, J. Fonlupt and D. Naddef: The traveling salesman problem on a graph and some related integer polyhedra, Mathematical Programming 33 (1985), 1–27.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    J. Edmonds: The Chinese postman’s problem, Bulletin of the Oper-ations Research Society of America 13 (1965), 373.Google Scholar
  11. [11]
    J. Edmonds: Submodular functions, matroids and certain polyhedra, in: Combinatorial Structures and Their Applications; Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications 1969 (R. Guy, H. Hanani, N. Sauer, J. Schönheim, eds.), Gordon and Breach New York 1970, 69–87.Google Scholar
  12. [12]
    J. Edmonds and E. L. Johnson: Matching, Euler tours and the Chinese postman, Mathematical Programming 5 (1973), 88–124.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    A. Frank: Conservative weightings and ear-decompositions of graphs, Combinatorica 13 (1993), 65–81.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    A. Frank: Connections in Combinatorial Optimization, Oxford University Press 2011.zbMATHGoogle Scholar
  15. [15]
    H. N. Gabow: Implementation of algorithms for maximum matching on nonbipartite graphs, Ph.D. thesis, Department of Computer Science, Stanford University 1973.Google Scholar
  16. [16]
    D. Gamarnik, M. Lewenstein and M. Sviridenko: An improved upper bound for the TSP in cubic 3-edge-connected graphs, Operations Research Letters 33 (2005), 467–474.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    M. R. Garey, D. S. Johnson and R. E. Tarjan: The planar Hamiltonian circuit problem is NP-complete, SIAM Journal on Computing 5 (1976), 704–714.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    J. A. Hoogeveen: Analysis of Christofides’ heuristic: some paths are more difficult than cycles, Operations Research Letters 10 (1991), 291–295.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    S. Khuller and U. Vishkin: Biconnectivity approximations and graph carvings, Journal of the ACM 41 (1994), 214–235.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    M. Lorea: Hypergraphes et matroïdes, Cahiers du Centre ďÉtudes de Recherche Opérationelle 17 (1975), 289–291.zbMATHMathSciNetGoogle Scholar
  21. [21]
    L. Lovász: A generalization of König’s theorem, Acta Mathematica Academiae Scientiarum Hungaricae 21 (1970), 443–446.CrossRefzbMATHMathSciNetGoogle Scholar
  22. [22]
    L. Lovász: 2-matchings and 2-covers of hypergraphs, Acta Mathematica Academiae Scientiarum Hungaricae 26 (1975), 433–444.CrossRefzbMATHMathSciNetGoogle Scholar
  23. [23]
    L. Lovász and M. D. Plummer: Matching Theory, Akadémiai Kiadó, Budapest 1986, and North-Holland, Amsterdam 1986.zbMATHGoogle Scholar
  24. [24]
    T. Mömke and O. Svensson: Approximating graphic TSP by matchings, Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (2011), 560–569.Google Scholar
  25. [25]
    C. L. Monma, B. S. Munson and W. R. Pulleyblank: Minimumweight two-connected spanning networks, Mathematical Programming 46 (1990), 153–171.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    M. Mucha: 13/9-approximation for graphic TSP, Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science (2012), 30–41.Google Scholar
  27. [27]
    S. Oveis Gharan, A. Saberi and M. Singh: A randomized rounding approach to the traveling salesman problem, Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (2011), 550–559.Google Scholar
  28. [28]
    C. H. Papadimitriou and M. Yannakakis: The traveling salesman problem with distances one and two, Mathematics of Operations Research 18 (1993), 1–12.CrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    R. Rado: A theorem on independence relations, Quarterly Journal of Mathematics 13 (1942), 83–89.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [30]
    A. Sebö: Eight fifth approximation for TSP paths, in: Integer Programming and Combinatorial Optimization; Proceedings of the 16th IPCO Conference; LNCS 7801 (J. Correa, M.X. Goemans, eds.), Springer Berlin 2013, 362–374.CrossRefGoogle Scholar
  31. [31]
    J. Vygen: New approximation algorithms for the TSP, OPTIMA 90 (2012), 1–12.Google Scholar
  32. [32]
    H. Whitney: Non-separable and planar graphs, Transactions of the American Mathematical Society 34 (1932), 339–362.CrossRefMathSciNetGoogle Scholar
  33. [33]
    L. A. Wolsey: Heuristic analysis, linear programming and branch and bound, Mathematical Programming Study 13 (1980), 121–134.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire G-SCOPCNRS — Univ. Grenoble AlpesGrenobleFrance
  2. 2.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations