On the Hardness of Reoptimization

  • Hans-Joachim Böckenhauer
  • Juraj Hromkovič
  • Tobias Mömke
  • Peter Widmayer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4910)

Abstract

We consider the following reoptimization scenario: Given an instance of an optimization problem together with an optimal solution, we want to find a high-quality solution for a locally modified instance. The naturally arising question is whether the knowledge of an optimal solution to the unaltered instance can help in solving the locally modified instance. In this paper, we survey some partial answers to this questions: Using some variants of the traveling salesman problem and the Steiner tree problem as examples, we show that the answer to this question depends on the considered problem and the type of local modification and can be totally different: For instance, for some reoptimization variant of the metric TSP, we get a 1.4-approximation improving on the best known approximation ratio of 1.5 for the classical metric TSP. For the Steiner tree problem on graphs with bounded cost function, which is APX-hard in its classical formulation, we even obtain a PTAS for the reoptimization variant. On the other hand, for a variant of TSP, where some vertices have to be visited before a prescribed deadline, we are able to show that the reoptimization problem is exactly as hard to approximate as the original problem.

Keywords

reoptimization approximation algorithms inapproximability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Archetti, C., Bertazzi, L., Speranza, M.G.: Reoptimizing the traveling salesman problem. Networks 42, 154–159 (2003)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ausiello, G., Escoffier, B., Monnot, J., Paschos, V.Th.: Reoptimization of minimum and maximum traveling salesman’s tours. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 196–207. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Bern, M.W., Plassmann, P.E.: The Steiner problem with edge lengths 1 and 2. Information Processing Letters 32(4), 171–176 (1989)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Böckenhauer, H.-J., Forlizzi, L., Hromkovič, J., Kneis, J., Kupke, J., Proietti, G., Widmayer, P.: Reusing optimal TSP solutions for locally modified input instances (extended abstract). In: IFIP TCS 2006. Proc. of the 4th IFIP International Conference on Theoretical Computer Science, pp. 251–270. Springer, Norwell (2006)CrossRefGoogle Scholar
  5. 5.
    Böckenhauer, H.-J., Forlizzi, L., Hromkovič, J., Kneis, J., Kupke, J., Proietti, G., Widmayer, P.: On the approximability of TSP on local modifications of optimally solved instances. Algorithmic Operations Research 2(2), 83–93 (2007)MathSciNetGoogle Scholar
  6. 6.
    Böckenhauer, H.-J., Hromkovič, J., Klasing, R., Seibert, S., Unger, W.: Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem (extended abstract). In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds.) CIAC 2000. LNCS, vol. 1767, pp. 72–86. Springer, Heidelberg (2000)Google Scholar
  7. 7.
    Böckenhauer, H.-J., Hromkovič, J., Klasing, R., Seibert, S., Unger, W.: Approximation algorithms for TSP with sharpened triangle inequality. Information Processing Letters 75, 133–138 (2000)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Böckenhauer, H.-J., Hromkovič, J., Klasing, R., Seibert, S., Unger, W.: An improved lower bound on the approximability of metric TSP and approximation algorithms for the TSP with sharpened triangle inequality (extended abstract). In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 382–394. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Böckenhauer, H.-J., Hromkovič, J., Klasing, R., Seibert, S., Unger, W.: Towards the notion of stability of approximation for hard optimization tasks and the traveling salesman problem. Theoretical Computer Science 285, 3–24 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Böckenhauer, H.-J., Hromkovič, J., Kneis, J., Kupke, J.: On the parameterized approximability of TSP with deadlines. Theory of Computing Systems (to appear)Google Scholar
  11. 11.
    Böckenhauer, H.-J., Hromkovič, J., Kneis, J., Kupke, J.: On the approximation hardness of some generalizations of TSP. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 184–195. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Böckenhauer, H.-J., Hromkovič, J., Královič, R., Mömke, T., Rossmanith, P.: Reoptimization of Steiner trees: changing the terminal set (submitted)Google Scholar
  13. 13.
    Böckenhauer, H.-J., Seibert, S.: Improved lower bounds on the approximability of the traveling salesman problem. RAIRO Theoretical Informatics and Applications 34, 213–255 (2000)MATHCrossRefGoogle Scholar
  14. 14.
    Christofides, N.: Worst-case analysis of a new heuristic for the travelling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh (1976)Google Scholar
  15. 15.
    Cordeau, J.-F., Desaulniers, G., Desrosiers, J., Solomon, M.M., Soumis, F.: VRP with time windows. In: Toth, P., Vigo, D. (eds.) The Vehicle Routing Problem, SIAM 2001, pp. 157–193 (2001)Google Scholar
  16. 16.
    Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1, 195–207 (1971/72)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Forlizzi, L., Hromkovič, J., Proietti, G., Seibert, S.: On the stability of approximation for Hamiltonian path problems. Algorithmic Operations Research 1(1), 31–45 (2006)MATHMathSciNetGoogle Scholar
  18. 18.
    Garey, M., Johnson, D.: Computers and Intractability. W. H. Freeman and Co., New York (1979)MATHGoogle Scholar
  19. 19.
    Greenberg, H.: An annotated bibliography for post-solution analysis in mixed integer and combinatorial optimization. In: Woodruff, D.L. (ed.) Advances in Computational and Stochastic Optimization, Logic Programming, and Heuristic Search, pp. 97–148. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  20. 20.
    Goldreich, O.: Bravely, moderately - A common theme in four recent works. In: SIGACT News, vol. 37, pp. 31–46. ACM, New York (2006)Google Scholar
  21. 21.
    Guttmann-Beck, N., Hassin, R., Khuller, S., Raghavachari, B.: Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem. Algorithmica 28, 422–437 (2000)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hoogeveen, J.A.: Analysis of Christofides’ heuristic: Some paths are more difficult than cycles. Operations Research Letters 10, 178–193 (1978)MathSciNetGoogle Scholar
  23. 23.
    Hromkovič, J.: Stability of approximation algorithms for hard optimization problems. In: Bartosek, M., Tel, G., Pavelka, J. (eds.) SOFSEM 1999. LNCS, vol. 1725, pp. 29–47. Springer, Heidelberg (1999)Google Scholar
  24. 24.
    Hromkovič, J.: Algorithmics for Hard Problems. Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics. Springer, Heidelberg (2003)Google Scholar
  25. 25.
    Libura, M.: Sensitivity analysis for minimum Hamiltonian path and traveling salesman problems. Discrete Applied Mathematics 30, 197–211 (1991)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Libura, M., van der Poort, E.S., Sierksma, G., van der Veen, J.A.A.: Stability aspects of the traveling salesman problem based on k-best solutions. Discrete Applied Mathematics 87, 159–185 (1998)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Mölle, D., Richter, S., Rossmanith, P.: A faster algorithm for the Steiner tree problem. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 561–570. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  28. 28.
    Prömel, H.J., Steger, A.: The Steiner Tree Problem. Friedr. Vieweg & Sohn, Braunschweig (2002)MATHGoogle Scholar
  29. 29.
    Papadimitriou, Ch., Steiglitz, K.: Some examples of difficult traveling salesman problems. Operations Research 26, 434–443 (1978)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. In: Proc. of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 770–779. ACM, New York (2000)Google Scholar
  31. 31.
    Sotskov, Y.N., Leontev, V.K., Gordeev, E.N.: Some concepts of stability analysis in combinatorial optimization. Discrete Appl. Math. 58, 169–190 (1995)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Van Hoesel, S., Wagelmans, A.: On the complexity of postoptimality analysis of 0/1 programs. Discrete Applied Mathematics 91, 251–263 (1999)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Hans-Joachim Böckenhauer
    • 1
  • Juraj Hromkovič
    • 1
  • Tobias Mömke
    • 1
  • Peter Widmayer
    • 1
  1. 1.Department of Computer Science, ETH ZurichSwitzerland

Personalised recommendations