Advertisement

A Coordinatewise Domain Scaling Algorithm for M-convex Function Minimization

  • Akihisa Tamura
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)

Abstract

We present a polynomial time domain scaling algorithm for the minimization of an M-convex function. M-convex functions are non-linear discrete functions with (poly)matroid structures, which are being recognized to play a fundamental role in tractable cases of discrete optimization. The novel idea of the algorithm is to use an individual scaling factor for each coordinate.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cunningham, W. H. and Frank, A.: A Primal-dual Algorithm for Submodular Flows. Math. Oper. Res. 10 (1985) 251–262MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Danilov, V., Koshevoy, G., Murota, K.: Discrete Convexity and Equilibria in Economies with Indivisible Goods and Money. Math. Social Sci. 41 (2001) 251–273MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Dress, A. W. M., Wenzel, W.: Valuated Matroid: A New Look at the Greedy Algorithm. Appl. Math. Lett. 3 (1990) 33–35MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Dress, A. W. M., Wenzel, W.: Valuated Matroids. Adv. Math. 93 (1992) 214–250MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Edmonds, J.: Submodular Functions, Matroids and Certain Polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.): Combinatorial Structures and Their Applications. Gordon and Breach, New York (1970) 69–87Google Scholar
  6. 6.
    Edmonds, J., Giles, R.: A Min-max Relation for Submodular Functions on Graphs. Ann. Discrete Math. 1 (1977) 185–204MathSciNetCrossRefGoogle Scholar
  7. 7.
    Frank, A.: Generalized Polymatroids. In: Hajnal, A., Lovász, L., Sós, V. T. (eds.): Finite and Infinite Sets, I. North-Holland, Amsterdam (1984) 285–294Google Scholar
  8. 8.
    Frank, A., Tardos, É.: Generalized Polymatroids and Submodular Flows. Math. Program. 42 (1988) 489–563MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Fujishige, S.: Lexicographically Optimal Base of a Polymatroid with Respect to a Weight Vector. Math. Oper. Res. 5 (1980) 186–196MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fujishige, S.: Submodular Functions and Optimization. Annals of Discrete Mathematics 47, North-Holland, Amsterdam (1991)Google Scholar
  11. 11.
    Groenevelt, H.: Two Algorithms for Maximizing a Separable Concave Function over a Polymatroid Feasible Region. European J. Oper. Res. 54 (1991) 227–236zbMATHCrossRefGoogle Scholar
  12. 12.
    Hochbaum, D. S.: Lower and Upper Bounds for the Allocation Problem and Other Nonlinear Optimization Problems. Math. Oper. Res. 19 (1994) 390–409MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Iwata, S.: A Fully Combinatorial Algorithm for Submodular Function Minimization. J. Combin. Theory Ser. B (to appear)Google Scholar
  14. 14.
    Iwata, S., Shigeno, M.: Conjugate Scaling Algorithm for Fenchel-type Duality in Discrete Convex Optimization. SIAM J. Optim. (to appear)Google Scholar
  15. 15.
    Iwata, S., Fleischer, L., Fujishige, S.: A Combinatorial Strongly Polynomial Algorithm for Minimizing Submodular Functions. J. ACM 48 (2001) 761–777MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lovász, L.: Submodular Functions and Convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.): Mathematical Programming — The State of the Art. Springer-Verlag, Berlin (1983) 235–257CrossRefGoogle Scholar
  17. 17.
    Moriguchi, S., Murota, K., Shioura, A.: Minimization of an M-convex Function with a Scaling Technique (in Japanese). IPSJ SIG Notes 2001-AL-76 (2001) 27–34Google Scholar
  18. 18.
    Moriguchi, S., Murota, K., Shioura, A.: Scaling Algorithms for M-convex Function Minimization. IEICE Trans. Fund. (to appear)Google Scholar
  19. 19.
    Murota, K.: Convexity and Steinitz’s Exchange Property. Adv. Math. 124 (1996) 272–311MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Murota, K.: Discrete Convex Analysis. Math. Program. 83 (1998) 313–371MathSciNetzbMATHGoogle Scholar
  21. 21.
    Murota, K.: Submodular Flow Problem with a Nonseparable Cost Function. Combinatorica 19 (1999) 87–109MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Murota, K.: Matrices and Matroids for Systems Analysis. Springer-Verlag, Berlin (2000)zbMATHGoogle Scholar
  23. 23.
    Murota, K.: Discrete Convex Analysis (in Japanese). Kyoritsu Publ. Co., Tokyo (2001)Google Scholar
  24. 24.
    Murota, K., Shioura, A.: Quasi M-convex and L-convex functions: Quasi-convexity in discrete optimization. Discrete Appl. Math. (to appear)Google Scholar
  25. 25.
    Murota, K., Tamura, A.: New Characterizations of M-convex Functions and Their Applications to Economic Equilibrium Models with Indivisibilities. Discrete Appl. Math. (to appear)Google Scholar
  26. 26.
    Murota, K., Tamura, A.: Application of M-convex Submodular Flow Problem to Mathematical Economics. In: Eades, P., Takaoka, T. (eds.): Proceedings of 12th International Symposium on Algorithms and Computation, Lecture Notes in Computer Science, Vol. 2223. Springer-Verlag, Berlin Heidelberg New York (2001) 14–25CrossRefGoogle Scholar
  27. 27.
    Rockafellar, R. T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  28. 28.
    Schrijver, A.: A Combinatorial Algorithm Minimizing Submodular Functions in Strongly Polynomial Time. J. Combin. Theory Ser. B 80 (2000) 346–355MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Shioura, A.: Minimization of an M-convex Function. Discrete Appl. Math. 84 (1998) 215–220MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Akihisa Tamura
    • 1
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

Personalised recommendations