Skip to main content

An Improved Analysis of the Mömke-Svensson Algorithm for Graph-TSP on Subquartic Graphs

  • Conference paper
Algorithms - ESA 2014 (ESA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8737))

Included in the following conference series:

  • 2267 Accesses

Abstract

Recently, Mömke and Svensson presented a beautiful new approach for the traveling salesman problem on a graph metric (graph-TSP), which yielded a \(\frac{4}{3}\)-approximation guarantee on subcubic graphs as well as a substantial improvement over the \(\frac{3}{2}\)-approximation guarantee of Christofides’ algorithm on general graphs. The crux of their approach is to compute an upper bound on the minimum cost of a circulation in a particular network, C(G,T), where G is the input graph and T is a carefully chosen spanning tree. The cost of this circulation is directly related to the number of edges in a tour output by their algorithm. Mucha subsequently improved the analysis of the circulation cost, proving that Mömke and Svensson’s algorithm for graph-TSP has an approximation ratio of at most \(\frac{13}{9}\) on general graphs.

This analysis of the circulation is local, and vertices with degree four and five can contribute the most to its cost. Thus, hypothetically, there could exist a subquartic graph (a graph with degree at most four at each vertex) for which Mucha’s analysis of the Mömke-Svensson algorithm is tight. In this paper, we show that this is not the case and that Mömke and Svensson’s algorithm for graph-TSP has an approximation guarantee of at most \(\frac{46}{33}\) on subquartic graphs. To prove this, we present a different method to upper bound the minimum cost of a circulation on the network C(G,T). Our approximation guarantee actually holds for all graphs that have an optimal solution to a standard linear programming relaxation of graph-TSP with subquartic support.

Supported in part by LabEx PERSYVAL-Lab (ANR–11-LABX-0025).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. An, H.-C., Kleinberg, R., Shmoys, D.B.: Improving Christofides’ algorithm for the s-t-path TSP. In: Proceedings of the 44th Symposium on Theory of Computing, pp. 875–886. ACM (2012)

    Google Scholar 

  2. 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)

    Chapter  Google Scholar 

  3. Cornuéjols, G., Fonlupt, J., Naddef, D.: The traveling salesman problem on a graph and some related integer polyhedra. Mathematical Programming 33(1), 1–27 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  4. Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, DTIC Document (1976)

    Google Scholar 

  5. Gao, Z.: An LP-based-approximation algorithm for the s-t path graph traveling salesman problem. Operations Research Letters 41(6), 615–617 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gharan, S.O., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 550–559. IEEE (2011)

    Google Scholar 

  8. Held, M., Karp, R.M.: The traveling-salesman problem and minimum spanning trees. Operations Research 18(6), 1138–1162 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hoogeveen, J.A.: Analysis of Christofides’ heuristic: some paths are more difficult than cycles. Operations Research Letters 10(5), 291–295 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mömke, T., Svensson, O.: Approximating graphic TSP by matchings. In: IEEE 52nd Annual Symposium on Foundations of Computer Science, pp. 560–569 (2011)

    Google Scholar 

  11. 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 für Informatik (2012)

    Google Scholar 

  12. Naddef, D., Pulleyblank, W.R.: Matchings in regular graphs. Discrete Mathematics 34(3), 283–291 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sebő, A.: Eight-fifth approximation for the path TSP. In: Goemans, M., Correa, J. (eds.) IPCO 2013. LNCS, vol. 7801, pp. 362–374. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  14. 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. arXiv:1201.1870 (2012)

    Google Scholar 

  15. Vishnoi, N.K.: A permanent approach to the traveling salesman problem. In: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pp. 76–80. IEEE (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Newman, A. (2014). An Improved Analysis of the Mömke-Svensson Algorithm for Graph-TSP on Subquartic Graphs. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_61

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-44777-2_61

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44776-5

  • Online ISBN: 978-3-662-44777-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics