Application of M-Convex Submodular Flow Problem to Mathematical Economics

  • Kazuo Murota
  • Akihisa Tamura
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2223)


This paper shows an application of the M-convexs ubmodular flow problem to an economic model in which producers and consumers trade various indivisible commodities through a perfectly divisible commodity, money. We give an efficient algorithm to decide whether a competitive equilibrium exists or not, when cost functions of the producers are M♮-convex and utility functions of the consumers are M♮-concave and quasilinear in money. The algorithm consists of two phases: the first phase computes productions and consumptions in an equilibrium by solving an M-convexs ubmodular flow problem and the second finds an equilibrium price vector by solving a shortest path problem.


Competitive Equilibrium Short Path Problem Convex Polyhedron Price Vector Initial Endowment 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. Danilov, G. Koshevoy, and K. Murota, Discrete convexity and equilibria in economies with indivisible goods and money, Math. Social Sci. 41 (2001) 251–273.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    S. Fujishige and K. Murota, Notes on L-/M-convexfu nctions and the separation theorems, Math. Programming 88 (2000) 129–146.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S. Iwata and M. Shigeno, Conjugate scaling algorithm for Fenchel-type duality in discrete convexo ptimization, SIAM J. Optim., to appear.Google Scholar
  4. 4.
    K. Murota, Convexity and Steinitz’s exchange property, Adv. Math. 124 (1996) 272–311.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    K. Murota, Discrete convexan alysis, Math. Programming 83 (1998) 313–371.MathSciNetGoogle Scholar
  6. 6.
    K. Murota, Submodular flow problem with a nonseparable cost function, Combinatorica 19 (1999) 87–109.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    K. Murota, Discrete Convex Analysis(in Japanese) (Kyoritsu-Shuppan, Tokyo) to appear.Google Scholar
  8. 8.
    K. Murota and A. Shioura, M-convexfu nction on generalized polymatroid, Math. Oper. Res. 24 (1999) 95–105.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    K. Murota and A. Shioura, Extension of M-convexity and L-convexity to polyhedral convexf unctions, Adv. in Appl. Math. 25 (2000) 352–427.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    K. Murota and A. Shioura, Relationship of M-/L-convexfu nctions with discrete convexfu nctions by Miller and by Favati-Tardella, Discrete Appl. Math. to appear.Google Scholar
  11. 11.
    K. Murota and A. Tamura, New characterizations of M-convexfu nctions and their applications to economic equilibrium models with indivisibilities, Discrete Appl.Math., to appear.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Kazuo Murota
    • 1
  • Akihisa Tamura
    • 1
  1. 1.RIMSKyoto UniversityKyotoJapan

Personalised recommendations