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Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs

Authors Dániel Marx, Ario Salmasi, Anastasios Sidiropoulos



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Dániel Marx
Ario Salmasi
Anastasios Sidiropoulos

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Dániel Marx, Ario Salmasi, and Anastasios Sidiropoulos. Constant-Factor Approximations for Asymmetric TSP on Nearly-Embeddable Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 16:1-16:54, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.16

Abstract

In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a closed walk of minimum cost in a directed graph visiting every vertex. We consider the approximability of ATSP on topologically restricted graphs. It has been shown by Oveis Gharan and Saberi [SODA, 2011] that there exists polynomial-time constant-factor approximations on planar graphs and more generally graphs of constant orientable genus. This result was extended to non-orientable genus by Erickson and Sidiropoulos [SoCG, 2014]. We show that for any class of nearly-embeddable graphs, ATSP admits a polynomial-time constant-factor approximation. More precisely, we show that for any fixed non-negative k, there exist positive alpha and beta, such that ATSP on n-vertex k-nearly-embeddable graphs admits an alpha-approximation in time O(n^beta). The class of k-nearly-embeddable graphs contains graphs with at most k apices, k vortices of width at most k, and an underlying surface of either orientable or non-orientable genus at most k. Prior to our work, even the case of graphs with a single apex was open. Our algorithm combines tools from rounding the Held-Karp LP via thin trees with dynamic programming. We complement our upper bounds by showing that solving ATSP exactly on graphs of pathwidth k (and hence on k-nearly embeddable graphs) requires time n^{Omega(k)}, assuming the Exponential-Time Hypothesis (ETH). This is surprising in light of the fact that both TSP on undirected graphs and Minimum Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth.
Keywords
  • asymmetric TSP
  • approximation algorithms
  • nearly-embeddable graphs
  • Held-Karp LP
  • exponential time hypothesis

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References

  1. Nima Anari and Shayan Oveis Gharan. Effective-resistance-reducing flows, spectrally thin trees, and asymmetric tsp. In 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS, 2015. Google Scholar
  2. Arash Asadpour, Michel X Goemans, Aleksander Madry, Shayan Oveis Gharan, and Amin Saberi. An O(log n/log log n)-approximation Algorithm for the Asymmetric Traveling Salesman Problem. In SODA, volume 10, pages 379-389. SIAM, 2010. Google Scholar
  3. Markus Bläser. A new approximation algorithm for the asymmetric tsp with triangle inequality. ACM Transactions on Algorithms (TALG), 4(4):47, 2008. Google Scholar
  4. Moses Charikar, Michel X Goemans, and Howard Karloff. On the integrality ratio for asymmetric tsp. In Foundations of Computer Science, 2004. Proceedings. 45th Annual IEEE Symposium on, pages 101-107. IEEE, 2004. Google Scholar
  5. Jianer Chen, Xiuzhen Huang, Iyad A Kanj, and Ge Xia. Strong computational lower bounds via parameterized complexity. Journal of Computer and System Sciences, 72(8):1346-1367, 2006. Google Scholar
  6. Holger Dell and Dániel Marx. Kernelization of packing problems. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 68-81. SIAM, 2012. Google Scholar
  7. Reinhard Diestel. Graph theory graduate texts in mathematics; 173. Springer-Verlag Berlin and Heidelberg GmbH &amp, 2000. Google Scholar
  8. Jeff Erickson and Anastasios Sidiropoulos. A near-optimal approximation algorithm for asymmetric tsp on embedded graphs. In Proceedings of the thirtieth annual symposium on Computational geometry, page 130. ACM, 2014. Google Scholar
  9. Uriel Feige and Mohit Singh. Improved approximation ratios for traveling salesperson tours and paths in directed graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 104-118. Springer, 2007. Google Scholar
  10. Fedor V Fomin, Petr A Golovach, Daniel Lokshtanov, and Saket Saurabh. Almost optimal lower bounds for problems parameterized by clique-width. SIAM Journal on Computing, 43(5):1541-1563, 2014. Google Scholar
  11. Alan M Frieze and Giulia Galbiati. On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks, 12(1):23-39, 1982. Google Scholar
  12. Alan M. Frieze, Giulia Galbiati, and Francesco Maffioli. On the worst-case performance of some algorithms for the asymmetric traveling salesman problem. Networks, 12(1):23-39, 1982. URL: http://dx.doi.org/10.1002/net.3230120103.
  13. Shayan Oveis Gharan and Amin Saberi. The asymmetric traveling salesman problem on graphs with bounded genus. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 967-975. SIAM, 2011. Google Scholar
  14. F. Harary. Graph Theory. Addison-Wesley Series in Mathematics. Perseus Books, 1994. Google Scholar
  15. Michael Held and Richard Karp. The traveling salesman problem and minimum spanning trees. Operations Research, 18:1138-1162, 1970. Google Scholar
  16. Michael Held and Richard M Karp. The traveling-salesman problem and minimum spanning trees. Operations Research, 18(6):1138-1162, 1970. Google Scholar
  17. Klaus Jansen, Stefan Kratsch, Dániel Marx, and Ildikó Schlotter. Bin packing with fixed number of bins revisited. Journal of Computer and System Sciences, 79(1):39-49, 2013. Google Scholar
  18. Haim Kaplan, Moshe Lewenstein, Nira Shafrir, and Maxim Sviridenko. Approximation algorithms for asymmetric tsp by decomposing directed regular multigraphs. Journal of the ACM (JACM), 52(4):602-626, 2005. Google Scholar
  19. Ken-ichi Kawarabayashi and Bojan Mohar. Some recent progress and applications in graph minor theory. Graphs and Combinatorics, 23(1):1-46, 2007. Google Scholar
  20. László Lovász. Graph minor theory. Bulletin of the American Mathematical Society, 43(1):75-86, 2006. Google Scholar
  21. A. Malnič and B. Mohar. Generating locally cyclic triangulations of surfaces. Journal of Combinatorial Theory, Series B, 56(2):147-164, 1992. Google Scholar
  22. Carsten Thomassen. Embeddings of graphs with no short noncontractible cycles. Journal of Combinatorial Theory, Series B, 48(2):155-177, 1990. Google Scholar
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