International Workshop on Approximation and Online Algorithms

Approximation and Online Algorithms pp 25-34 | Cite as

Maximum ATSP with Weights Zero and One via Half-Edges

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9499)

Abstract

We present a fast combinatorial 3 / 4-approximation algorithm for the maximum asymmetric TSP with weights zero and one. The approximation factor of this algorithm matches the currently best one given by Bläser in 2004 and based on linear programming. Our algorithm first computes a maximum size matching and a maximum weight cycle cover without certain cycles of length two but possibly with half-edges - a half-edge of a given edge e is informally speaking a half of e that contains one of the endpoints of e. Then from the computed matching and cycle cover it extracts a set of paths, whose weight is large enough to be able to construct a traveling salesman tour with the claimed guarantee.

References

  1. 1.
    Bläser, M.: An 8/13-approximation algorithm for the asymmetric maximum TSP. J. Algorithms 50(1), 23–48 (2004)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bläser, M.: A 3/4-approximation algorithm for maximum ATSP with weights zero and one. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 61–71. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  3. 3.
    Bläser, M., Manthey, B.: Two approximation algorithms for 3-cycle covers. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, p. 40. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  4. 4.
    Bläser, M., Siebert, B.: Computing cycle covers without short cycles. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 369–379. Springer, Heidelberg (2001) Google Scholar
  5. 5.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for finding a maximum weight Hamiltonian circuit. Oper. Res. 27(4), 799–809 (1979)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.: Approximation algorithms for asymmetric tsp by decomposing directed regular multigraphs. J. ACM 52(4), 602–626 (2005). Preliminary version appeared in FOCS’03MathSciNetCrossRefGoogle Scholar
  7. 7.
    Karpinski, M., Schmied, R.: Improved Inapproximability results for the shortest superstring and related problems. In: CATS, pp. 27–36 (2013)Google Scholar
  8. 8.
    Kosaraju, S.R., Park, J.K., Stein, C.: Long tours and short superstrings (preliminary version). In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 166–177 (1994)Google Scholar
  9. 9.
    Kowalik, L., Mucha, M.: Deterministic 7/8-approximation for the metric maximum tsp. Theor. Comput. Sci. 410(47–49), 5000–5009 (2009)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Kowalik, L., Mucha, M.: 35/44-approximation for asymmetric maximum tsp with triangle inequality. Algorithmica 59(2), 240–255 (2011)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Lewenstein, M., Sviridenko, M.: A 5/8 approximation algorithm for the maximum asymmetric tsp. SIAM J. Discrete Math. 17(2), 237–248 (2003)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Lovasz, L., Plummer, M.D.: Matching Theory (1986)Google Scholar
  13. 13.
    Paluch, K., Mucha, M., Madry, A.: A 7/9 - approximation algorithm for the maximum traveling salesman problem. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) Approximation, Randomization, and Combinatorial Optimization. LNCS, vol. 5687, pp. 298–311. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  14. 14.
    Paluch, K.E., Elbassioni, K.M., van Zuylen, A.: Simpler approximation of the maximum asymmetric traveling salesman problem. In: Proceedings of the 29th Symposium on Theoretical Aspects of Computer Science, STACS 2012, Leibniz International Proceedings of Informatics 14, pp. 501–506 (2012)Google Scholar
  15. 15.
    Paluch, K.: Better Approximation Algorithms for Maximum Asymmetric Traveling Salesman and Shortest Superstring. CoRR abs/1401.3670 (2014)Google Scholar
  16. 16.
    Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Math. Oper. Res. 18, 1–11 (1993)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Vishwanathan, S.: An approximation algorithm for the asymmetric travelling salesman problem with distances one and two. Inform. Proc. Lett. 44, 297–302 (1992)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Wroclaw UniversityWroclawPoland

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