Abstract
In this extended abstract, we survey some of the recent results on approximating the traveling salesman problem on graphic metrics.
We start by briefly explaining the algorithm of Oveis Gharan et al. [1] that has strong similarities to Christofides’ famous 3/2-approximation algorithm. We then explain the main ideas behind an alternative approach introduced by Mömke and the author [2]. The new ingredient in our approach is that it allows for the removal of certain edges while simultaneously yielding a connected, Eulerian graph, which in turn leads to a decreased cost. We also overview the exciting developments for TSP on graphic metrics that rapidly followed: an improved analysis of our algorithm by Mucha [3] yielding an approximation guarantee of 1.44, and the recent developments by Sebö and Vygen [3] who gave a 1.4-approximation algorithm.
Finally, we point out some interesting open problems where our techniques currently fall short of applying to more general metrics.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Oveis Gharan, S., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: [18], pp. 550–559
Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: [18], pp. 560–569
Mucha, M.: 13/9-approximation for graphic TSP. In: Dürr, C., Wilke, T. (eds.) STACS. LIPIcs, vol. 14, pp. 30–41. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)
Lampis, M.: Improved inapproximability for TSP. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX/RANDOM 2012. LNCS, vol. 7408, pp. 243–253. Springer, Heidelberg (2012)
Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University (1976)
Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM 45(5), 753–782 (1998)
Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing 28(4), 1298–1309 (1999)
Arora, S., Grigni, M., Karger, D.R., Klein, P.N., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Karloff, H.J. (ed.) SODA, pp. 33–41. ACM/SIAM (1998)
Grigni, M., Koutsoupias, E., Papadimitriou, C.H.: An approximation scheme for planar graph TSP. In: Proc. 36th Annual Symposium on Foundations of Computer Science (FOCS 1995), pp. 640–645. IEEE Computer Society (1995)
Goemans, M.X.: Worst-case comparison of valid inequalities for the TSP. Mathematics and Statistics 69(1), 335–349 (1995)
Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem. In: Charikar, M. (ed.) SODA, pp. 379–389. SIAM (2010)
Frederickson, G.N., Jájá, J.: On the relationship between the biconnectivity augmentation and travelling salesman problems. Theoretical Computer Science 19(2), 189–201 (1982)
Monma, C.L., Munson, B.S., Pulleyblank, W.R.: Minimum-weight two-connected spanning networks. Mathematical Programming 46(1), 153–171 (1990)
Boyd, S., Sitters, R., van der Ster, S., Stougie, L.: TSP on cubic and subcubic graphs. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 65–77. Springer, Heidelberg (2011)
Gamarnik, D., Lewenstein, M., Sviridenko, M.: An improved upper bound for the TSP in cubic 3-edge-connected graphs. Operations Research Letters 33(5), 467–474 (2005)
Sebő, A., Vygen, J.: Shorter tours by nicer ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs. CoRR abs/1201.1870 (2012)
Vygen, J.: New approximation algorithms for the TSP. In: OPTIMA, vol. 90, http://www.mathopt.org/Optima-Issues/optima90.pdf
Ostrovsky, R. (ed.): IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25. IEEE (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Svensson, O. (2013). Overview of New Approaches for Approximating TSP. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-45043-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-45042-6
Online ISBN: 978-3-642-45043-3
eBook Packages: Computer ScienceComputer Science (R0)