Skip to main content
SpringerLink
Log in

Constant factor approximation for ATSP with two edge weights

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

We give a constant factor approximation algorithm for the Asymmetric Traveling Salesman Problem on shortest path metrics of directed graphs with two different edge weights. For the case of unit edge weights, the first constant factor approximation was given recently by Svensson. This was accomplished by introducing an easier problem called Local-Connectivity ATSP and showing that a good solution to this problem can be used to obtain a constant factor approximation for ATSP. In this paper, we solve Local-Connectivity ATSP for two different edge weights. The solution is based on a flow decomposition theorem for solutions of the Held–Karp relaxation, which may be of independent interest.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. An estimation algorithm is a polynomial-time algorithm for approximating/estimating the optimal value without necessarily finding a solution to the problem.

  2. For ATSP, we can think of a node-weighted graph as an edge-weighted graph where the weight of an edge (u, v) equals the node weight of u.

  3. That is, the maximum flow value from s to any proper subset \(T'\subsetneq T\) is smaller than \(c(\delta ^+(s))\).

  4. Note that \(s \notin A\) since \(D_f\) contains a path from \(t_k\) to s.

  5. Note that we decompose the vertex set \(V_i\), but with respect to the edge set \(E(G_i^{\mathrm {aux}})\), not \(E(G[V_i])\).

  6. To obtain exactly 1 / 2, we might need to break an edge up into two copies, dividing its \(x^\star _{\mathrm {sp}}\)-value between them appropriately, and include one copy in \(X_i^-\) but not the other; we omit this for simplicity of notation, and assume there is such an edge set with exactly \(x^\star _{\mathrm {sp}}(X_i^-)= 1/2\).

  7. Again, we might need to split some edges into two copies.

  8. It is violated unless \(u_i = v_i\).

  9. Note that this walk may exit \(U_i\), but it will stay inside \(V_i\).

  10. Map the path given by Fact 3.11 from \(G_{\mathrm {sp}}\) to G.

References

  1. Anari, N., Oveis Gharan, S.: Effective-resistance-reducing flows and asymmetric TSP. In: Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE (2015)

  2. Arora, S., Grigni, M., Karger, D.R., Klein, P.N., Woloszyn, A.: A polynomial-time approximation scheme for weighted planar graph TSP. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1998, pp. 33–41 (1998)

  3. Asadpour, A., Goemans, M.X., Madry, A., Oveis Gharan, S., Saberi, A.: An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 379–389 (2010)

    Chapter  Google Scholar 

  4. Berman, P., Karpinski, M.: 8/7-approximation algorithm for (1, 2)-TSP. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 641–648 (2006)

  5. Bläser, M.: A 3/4-approximation algorithm for maximum ATSP with weights zero and one. In: Approximation, Randomization, and Combinatorial Optimization, Algorithms and Techniques, 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX, pp. 61–71 (2004)

    Chapter  Google Scholar 

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

  7. Erickson, J., Sidiropoulos, A.: A near-optimal approximation algorithm for asymmetric TSP on embedded graphs. In: 30th Annual Symposium on Computational Geometry (SOCG), pp. 130 (2014)

  8. Frieze, A.M., Galbiati, G., Maffioli, F.: On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks 12(1), 23–39 (1982)

    Article  MathSciNet  Google Scholar 

  9. Oveis Gharan, S., Saberi, A.: The asymmetric traveling salesman problem on graphs with bounded genus. In: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pp. 967–975. SIAM (2011)

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

  11. Grigni, M., Koutsoupias, E., Papadimitriou, C.H.: An approximation scheme for planar graph TSP. In: 36th Annual Symposium on Foundations of Computer Science, FOCS 1995, pp. 640–645 (1995)

  12. Karpinski, M., Lampis, M., Schmied, R.: New inapproximability bounds for TSP. J. Comput. Syst. Sci. 81(8), 1665–1677 (2015)

    Article  MathSciNet  Google Scholar 

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

  14. Mucha, M.: 13/9-approximation for graphic TSP. In: 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, pp. 30–41 (2012)

  15. Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res. 18(1), 1–11 (1993)

    Article  MathSciNet  Google Scholar 

  16. 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(5), 597–629 (2014)

  17. Svensson, O.: Approximating ATSP by relaxing connectivity. In: Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2015)

  18. Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)

Download references

Author information

Authors and Affiliations

  1. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Ola Svensson & Jakub Tarnawski

  2. London School of Economics, London, UK

    László A. Végh

Authors
  1. Ola Svensson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jakub Tarnawski
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. László A. Végh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jakub Tarnawski.

Additional information

O. Svensson and J. Tarnawski: supported by ERC Starting Grant 335288-OptApprox; László A. Végh: supported by EPSRC First Grant EP/M02797X/1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Svensson, O., Tarnawski, J. & Végh, L.A. Constant factor approximation for ATSP with two edge weights. Math. Program. 172, 371–397 (2018). https://doi.org/10.1007/s10107-017-1195-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-017-1195-7

Mathematics Subject Classification

Search

Navigation