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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
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.
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), 875–886
F. Barahona and M. Conforti: A construction for binary matroids, Discrete Mathematics 66 (1987), 213–218.
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.
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.
J. Cheriyan, A. Sebő 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.
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.
W. J. Cook: In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation, Princeton University Press 2012.
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.
J. Edmonds: The Chinese postman’s problem, Bulletin of the Operations Research Society of America 13 (1965), 3–3.
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.
J. Edmonds and E. L. Johnson: Matching, Euler tours and the Chinese postman, Mathematical Programming 5 (1973), 88–124.
A. Frank: Conservative weightings and ear-decompositions of graphs, Combinatorica 13 (1993), 65–81.
A. Frank: Connections in Combinatorial Optimization, Oxford University Press 2011.
H. N. Gabow: Implementation of algorithms for maximum matching on nonbipartite graphs, Ph.D. thesis, Department of Computer Science, Stanford University 1973.
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.
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.
J. A. Hoogeveen: Analysis of Christofides’ heuristic: some paths are more difficult than cycles, Operations Research Letters 10 (1991), 291–295.
S. Khuller and U. Vishkin: Biconnectivity approximations and graph carvings, Journal of the ACM 41 (1994), 214–235.
M. Lorea: Hypergraphes et matroïdes, Cahiers du Centre d’Études de Recherche Opérationelle 17 (1975), 289–291.
L. Lovász: A generalization of Kőnig’s theorem, Acta Mathematica Academiae Scientiarum Hungaricae 21 (1970), 443–446.
L. Lovász: 2-matchings and 2-covers of hypergraphs, Acta Mathematica Academiae Scientiarum Hungaricae 26 (1975), 433–444.
L. Lovász and M. D. Plummer: Matching Theory, Akadémiai Kiadó, Budapest 1986, and North-Holland, Amsterdam 1986.
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.
C. L. Monma, B. S. Munson and W. R. Pulleyblank: Minimum-weight twoconnected spanning networks, Mathematical Programming 46 (1990), 153–171.
M. Mucha: 13/9-approximation for graphic TSP, Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science (2012), 30–41.
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.
C. H. Papadimitriou and M. Yannakakis: The traveling salesman problem with distances one and two, Mathematics of Operations Research 18 (1993), 1–12.
R. Rado: A theorem on independence relations, Quarterly Journal of Mathematics 13 (1942), 83–89.
A. Sebő: 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.
J. Vygen: New approximation algorithms for the TSP, OPTIMA 90 (2012), 1–12.
H. Whitney: Non-separable and planar graphs, Transactions of the American Mathematical Society 34 (1932), 339–362.
L. A. Wolsey: Heuristic analysis, linear programming and branch and bound, Mathematical Programming Study 13 (1980), 121–134.
Supported by LabEx PERSYVAL-Lab (ANR-11-LABX-0025), and TEOMATRO (ANR-10-BLAN 0207).
This work was done while visiting Grenoble, Laboratoire G-SCOP. Support of Université Joseph Fourier is gratefully acknowledged.
About this article
Cite this article
Sebő, A., Vygen, J. 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. Combinatorica 34, 597–629 (2014). https://doi.org/10.1007/s00493-014-2960-3
Mathematics Subject Classification (2000)