Algorithmica

, Volume 67, Issue 1, pp 89–110 | Cite as

The Parameterized Complexity of Local Search for TSP, More Refined

  • Jiong Guo
  • Sepp Hartung
  • Rolf Niedermeier
  • Ondřej Suchý
Article

Abstract

We extend previous work on the parameterized complexity of local search for the Traveling Salesperson Problem (TSP). So far, its parameterized complexity has been investigated with respect to the distance measures (defining the local search area) “Edge Exchange” and “Max-Shift”. We perform studies with respect to the distance measures “Swap” and “r-Swap”, “Reversal” and “r-Reversal”, and “Edit”, achieving both fixed-parameter tractability and W[1]-hardness results. In particular, from the parameterized reduction showing W[1]-hardness we infer running time lower bounds (based on the exponential time hypothesis) for all corresponding distance measures. Moreover, we provide non-existence results for polynomial-size problem kernels and we show that some in general W[1]-hard problems turn fixed-parameter tractable when restricted to planar graphs.

Keywords

NP-hard problem Heuristics Problem kernel Fixed-parameter tractability W[1]-hardness Lower bounds based on ETH 

References

  1. 1.
    Arora, S.: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Balas, E.: New classes of efficiently solvable generalized traveling salesman problems. Ann. Oper. Res. 86, 529–558 (1999) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209, 1–45 (1998) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Kernelization: new upper and lower bound techniques. In: Proceedings of the 4th International Workshop on Parameterized and Exact Computation (IWPEC ’09). Lecture Notes in Computer Science, vol. 5917, pp. 17–37. Springer, Berlin (2009) CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Caprara, A.: Sorting by reversals is difficult. In: Proceedings of the 1st Conference on Research in Computational Molecular Biology (RECOMB ’97), pp. 75–83. ACM, New York (1997) Google Scholar
  7. 7.
    Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D.W., Kanj, I.A., Xia, G.: Tight lower bounds for certain parameterized NP-hard problems. Inf. Comput. 201(2), 216–231 (2005) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, Carnegie Mellon University Pittsburgh , Management Sciences Research Group (1976) Google Scholar
  9. 9.
    Deineko, V.G., Woeginger, G.J.: A study of exponential neighborhoods for the travelling salesman problem and for the quadratic assignment problem. Math. Program. 87(3), 519–542 (2000) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dorn, F., Penninkx, E., Bodlaender, H.L., Fomin, F.: Efficient exact algorithms on planar graphs: exploiting sphere cut decompositions. Algorithmica 58, 790–810 (2010) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) CrossRefGoogle Scholar
  12. 12.
    Fellows, M.R.: Towards fully multivariate algorithmics: some new results and directions in parameter ecology. In: Proceedings of the 20th International Workshop on Combinatorial Algorithms (IWOCA ’09). Lecture Notes in Computer Science, vol. 5874, pp. 2–10. Springer, Berlin (2009) CrossRefGoogle Scholar
  13. 13.
    Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Fellows, M.R., Fomin, F.V., Lokshtanov, D., Rosamond, F., Saurabh, S., Villanger, Y.: Local search: is brute-force avoidable? J. Comput. Syst. Sci. 78(3), 707–719 (2012) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Fertin, G., Labarre, A., Rusu, I., Tannier, É., Vialette, S.: Combinatorics of Genome Rearrangements. MIT Press, Cambridge (2009) MATHGoogle Scholar
  16. 16.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  17. 17.
    Fomin, F., Lokshtanov, D., Raman, V., Saurabh, S.: Fast local search algorithm for weighted feedback arc set in tournaments. In: Proceedings of the 24th Conference on Artificial Intelligence (AAAI ’10), pp. 65–70 (2010) Google Scholar
  18. 18.
    Garey, M., Johnson, D., Tarjan, R.: The planar Hamiltonian circuit problem is NP-complete. SIAM J. Comput. 5(4), 704–714 (1976) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Gaspers, S., Kim, E.J., Ordyniak, S., Saurabh, S., Szeider, S.: Don’t be strict in local search. In: Proceedings of the 26th Conference on Artificial Intelligence (AAAI ’12), pp. 486–492 (2012) Google Scholar
  20. 20.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007) CrossRefGoogle Scholar
  21. 21.
    Gutin, G., Punnen, A.: The Traveling Salesman Problem and Its Variations. Kluwer Academic, Norwell (2002) MATHGoogle Scholar
  22. 22.
    Gutin, G., Yeo, A., Zverovitch, A.: Exponential neighborhoods and domination analysis for the TSP. In: The Traveling Salesman Problem and Its Variations, pp. 223–256. Kluwer Academic, Norwell (2002). Chapter 1 Google Scholar
  23. 23.
    Hartung, S., Niedermeier, R.: Incremental list coloring of graphs, parameterized by conservation. In: Proceedings of the 7th Annual Conference on Theory and Applications of Models of Computation (TAMC ’10). Lecture Notes in Computer Science, vol. 6108, pp. 258–270. Springer, Berlin (2010) CrossRefGoogle Scholar
  24. 24.
    Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. J. Soc. Ind. Appl. Math. 10(1), 196–210 (1962) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Johnson, D.S., McGeoch, L.A.: The traveling salesman problem: a case study in local optimization. In: Local Search in Combinatorial Optimization, pp. 215–310 (1997) Google Scholar
  27. 27.
    Johnson, D.S., McGeoch, L.A.: Experimental analysis of heuristics for the STSP. In: The Traveling Salesman Problem and Its Variations, pp. 369–443. Kluwer Academic, Norwell (2004) CrossRefGoogle Scholar
  28. 28.
    Krokhin, A., Marx, D.: On the hardness of losing weight. ACM Trans. Algorithms 8(2), 19:1–19:18 (2012) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lawler, E., Lenstra, J., Kan, A., Shmoys, D.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, vol. 3. Wiley, New York (1985) MATHGoogle Scholar
  30. 30.
    Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bull. Eur. Assoc. Theor. Comput. Sci. 84, 41–71 (2011) MathSciNetGoogle Scholar
  31. 31.
    Lokshtanov, D., Misra, N., Saurabh, S.: Kernelization–preprocessing with a guarantee. In: The Multivariate Algorithmic Revolution and Beyond–Essays Dedicated to Michael R. Fellows. Lecture Notes in Computer Science, vol. 7370, pp. 130–161. Springer, Berlin (2012) Google Scholar
  32. 32.
    Marx, D.: Searching the k-change neighborhood for TSP is W[1]-hard. Oper. Res. Lett. 36(1), 31–36 (2008) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Marx, D.: Can you beat treewidth? Theory Comput. 6(1), 85–112 (2010) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Marx, D., Schlotter, I.: Stable assignment with couples: parameterized complexity and local search. Discrete Optim. 8(1), 25–40 (2011) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006) MATHCrossRefGoogle Scholar
  36. 36.
    Niedermeier, R.: Reflections on multivariate algorithmics and problem parameterization. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS ’10). LIPIcs, vol. 5, pp. 17–32 (2010). Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik Google Scholar
  37. 37.
    Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Comb. Theory, Ser. B 41, 92–114 (1986) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory, Ser. B 52(2), 153–190 (1991) MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Szeider, S.: The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optim. 8(1), 139–145 (2011) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Jiong Guo
    • 1
  • Sepp Hartung
    • 2
  • Rolf Niedermeier
    • 2
  • Ondřej Suchý
    • 2
  1. 1.Cluster of Excellence, Multimodal Computing and InteractionUniversität des SaarlandesSaarbrückenGermany
  2. 2.Institut für Softwaretechnik und Theoretische InformatikTU BerlinBerlinGermany

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