Advertisement

Approximation with a Fixed Number of Solutions of Some Biobjective Maximization Problems

  • Cristina Bazgan
  • Laurent Gourvès
  • Jérôme Monnot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7164)

Abstract

We investigate the problem of approximating the Pareto set of biobjective optimization problems with a given number of solutions. This task is relevant for two reasons: (i) Pareto sets are often computationally hard so approximation is a necessary tradeoff to allow polynomial time algorithms; (ii) limiting explicitly the size of the approximation allows the decision maker to control the expected accuracy of approximation and prevents him to be overwhelmed with too many alternatives. Our purpose is to exploit general properties that many well studied problems satisfy. We derive existence and constructive approximation results for the biobjective versions of Max Bisection, Max Partition, Max Set Splitting and Max Matching.

Keywords

Feasible Solution Polynomial Time Multiobjective Optimization Polynomial Time Approximation Scheme Constant Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alimonti, P.: Non-Oblivious Local Search for Graph and Hypergraph Coloring Problems. In: Nagl, M. (ed.) WG 1995. LNCS, vol. 1017, pp. 167–180. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  2. 2.
    Angel, E., Bampis, E., Gourvès, L.: Approximation algorithms for the bi-criteria weighted max-cut problem. Discrete Applied Mathematics 154(12), 1685–1692 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Angel, E., Bampis, E., Gourvès, L., Monnot, J.: (Non)-Approximability for the Multi-criteria TSP(1,2). In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 329–340. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Angel, E., Bampis, E., Kononov, A.: On the approximate tradeoff for bicriteria batching and parallel machine scheduling problems. Theoretical Computer Science 306(1-3), 319–338 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bazgan, C., Hugot, H., Vanderpooten, D.: Implementing an efficient fptas for the 0-1 multi-objective knapsack problem. European Journal of Operational Research 198(1), 47–56 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press (2009)Google Scholar
  7. 7.
    Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ehrgott, M.: Multicriteria optimization. LNEMS. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  9. 9.
    Erlebach, T., Kellerer, H., Pferschy, U.: Approximating multiobjective knapsack problems. Management Science 48(12), 1603–1612 (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of ACM 42(6), 1115–1145 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Halperin, E., Zwick, U.: A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems. Random Structure Algorithms 20(3), 382–402 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hansen, P.: Bicriteria path problems. In: Fandel, G., Gal, T. (eds.) Multiple Criteria Decision Making: Theory and Applications, pp. 109–127 (1980)Google Scholar
  13. 13.
    Hastad, J.: Some optimal inapproximability results. Journal of ACM 48(4), 798–859 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kann, V., Lagergren, J., Panconesi, A.: Approximability of maximum splitting of k-sets and some other apx-complete problems. Information Processing Letters 58(3), 105–110 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)Google Scholar
  16. 16.
    Manthey, B.: On Approximating Multi-Criteria TSP. In: Albers, S., Marion, J.-Y. (eds.) Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS 2009). LIPIcs, pp. 637–648 (2009)Google Scholar
  17. 17.
    Paluch, K., Mucha, M., Mądry, A.: A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 298–311. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS 2000), pp. 86–92 (2000)Google Scholar
  19. 19.
    Serafini, P.: Some considerations about computational complexity for multi objective combinatorial problems. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 294, pp. 222–232 (1986)Google Scholar
  20. 20.
    Stein, C., Wein, J.: On the existence of schedules that are near-optimal for both makespan and total weighted completion time. Operational Research Letters 21(3), 115–122 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Tsaggouris, G., Zaroliagis, C.: Multiobjective Optimization: Improved FPTAS for Shortest Paths and Non-linear Objectives with Applications. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 389–398. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Warburton, A.: Approximation of pareto-optima in multiple-objective shortest path problems. Operations Research 35(1), 70–79 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Woeginger, G.: A polynomial time approximation scheme for maximizing the minimum machine completion time. Operations Research Letters 20(4), 149–154 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Zhang, J., Yea, Y., Han, Q.: Improved approximations for max set splitting and max NAE SAT. Discrete Applied Mathematics 142(1-3), 133–149 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Zwick, U.: Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In: Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1998), pp. 201–210 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Cristina Bazgan
    • 1
    • 2
    • 3
  • Laurent Gourvès
    • 1
    • 2
  • Jérôme Monnot
    • 1
    • 2
  1. 1.LAMSADEUniversité Paris-DauphineParis Cedex 16France
  2. 2.CNRS, UMR 7243France
  3. 3.Institut Universitaire de FranceFrance

Personalised recommendations