A Polynomial-Time-Delay and Polynomial-Space Algorithm for Enumeration Problems in Multi-criteria Optimization

  • Yoshio Okamoto
  • Takeaki Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4835)

Abstract

We propose a polynomial-time-delay polynomial-space algorithm to enumerate all efficient extreme solutions of a multi-criteria minimum-cost spanning tree problem, while only the bi-criteria case was studied in the literature. The algorithm is based on the reverse search framework due to Avis & Fukuda. We also show that the same technique can be applied to the multi-criteria version of the minimum-cost basis problem in a (possibly degenerated) submodular system. As an ultimate generalization, we propose an algorithm to enumerate all efficient extreme solutions of a multi-criteria linear program. When the given linear program has no degeneracy, the algorithm runs in polynomial-time delay and polynomial space. To best of our knowledge, they are the first polynomial-time delay and polynomial-space algorithms for the problems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete Applied Mathematics 65, 21–46 (1996)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ehrgott, M.: On matroids with multiple objectives. Optimization 38, 73–84 (1996)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Heidelberg (2005)MATHGoogle Scholar
  4. 4.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, 2nd edn. Springer, Berlin New York (1993)MATHGoogle Scholar
  5. 5.
    Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V.: Generating all vertices of a polyhedron is hard. In: Proc. 17th SODA. Full version to appear in Discrete & Computational Geometry, pp. 758–765 (2006)Google Scholar
  6. 6.
    Nakano, S.-I., Uno, T.: Constant time generation of trees with specified diameter. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 33–45. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Papadimitriou, C., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proc. 41st FOCS, pp. 86–92 (2000)Google Scholar
  8. 8.
    Ulungu, E.L., Teghem, J.: The two phase method: an efficient procedure to solve bi-objective combinatorial optimization problems. Foundations of Computing and Decision Sciences 20, 149–165 (1995)MATHMathSciNetGoogle Scholar
  9. 9.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)Google Scholar
  10. 10.
    Zaroliagis, C.: Recent advances in multiobjective optimization. In: Lupanov, O.B., Kasim-Zade, O.M., Chaskin, A.V., Steinhöfel, K. (eds.) SAGA 2005. LNCS, vol. 3777, pp. 45–47. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Zitzler, E., Laumanns, M., Bleuler, S.: A tutorial on evolutionary multiobjective optimization. In: Gandibleux, X., Sevaux, M., Sörensen, K., T’kindt, V. (eds.) Metaheuristics for Multiobjective Optimisation, Lecture Notes in Economics and Mathematical Systems, vol. 535, pp. 3–38 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Yoshio Okamoto
    • 1
  • Takeaki Uno
    • 2
  1. 1.Department of Information and Computer Sciences, Toyohashi University of Technlology, Hibarigaoka 1–1, Tempaku, Toyohashi, Aichi, 441-8580Japan
  2. 2.National Institute of Informatics, Hitotsubashi 2–1–2, Chiyoda-ku, Tokyo, 101-8430Japan

Personalised recommendations