Methods of the Convex Cone Theory in the Feasibility Problem of Multicommodity Flow

  • Ya. R. Grinberg


The feasibility problem of multicommodity flow is reduced to finding out if a multidimensional vector determined by the network parameters belongs to a convex polyhedral cone determined by the set of paths in the network. It is shown that this representation of the feasibility problem makes it possible to formulate the feasibility criterion described in [1] in a different form. It is proved that this criterion is sufficient. The concepts of reference vectors and networks are defined, and they are used to describe a method for solving the feasibility problem for an arbitrary network represented by a complete graph.


multicommodity flow feasibility criterion polyhedral cone multivertex graph 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. B. Papernov, “Realizability of multicommodity flows,” in Discrete Optimization Studies (Nauka, Moscow, 1976), pp. 230–261 [in Russian].Google Scholar
  2. 2.
    L. R. Ford and D. R. Fulkerson, Flows in Networks (Princeton Univ. Press, Princeton, 1962; Mir, Moscow, 1966).zbMATHGoogle Scholar
  3. 3.
    T. C. Hu, Integer Programming and Network Flows (Addison–Wesley, Reading, Mass., 1970; Mir, Moscow, 1974).Google Scholar
  4. 4.
    M. V. Lomonosov, “On system of flows in a network,” Probl. Inf. Transm. 14 (4), 280–290 (1978).MathSciNetzbMATHGoogle Scholar
  5. 5.
    B. N. Pshenichyi, Convex Analysis and Extremum Value Problems (Nauka, Moscow, 1980) [in Russian].Google Scholar
  6. 6.
    M. R. Davidson, Yu. E. Malashenko, N. M. Novikova, M. M. Smirnov, and G. V. Strogaya, Mathematical Statements of Problems of Recovery and Survivability for Multicommodity Networks (Vychisl. Tsentr, Ross. Akad. Nauk, Moscow, 1993) [in Russian].zbMATHGoogle Scholar
  7. 7.
    M. R. Davidson, “Stability of the lexicographic maximin problem of distributing flows in multicommodity networks,” Zh. Vychisl. Mat. Mat. Fiz. 35, 334–351 (1995).Google Scholar
  8. 8.
    Yu. E. Malashenko and N. M. Novikova, “Superconcurrent distribution of flows in multicommodity networks,” Discretn. Anal. Issled. Oper. 4 (2), 34–54 (1997).MathSciNetzbMATHGoogle Scholar
  9. 9.
    A. Schrijver, Theory of Linear and Integer Programming (Wiley, Chichester, 1986; Mir, Moscow, 1991).zbMATHGoogle Scholar
  10. 10.
    Ya. R. Grinberg, “Step graphs and their application to the organization of commodity flows in networks,” J. Comput. Syst. Sci. Int. 55, 222–231 (2016).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Kharkevich Institute for Information Transmission ProblemsRussian Academy of SciencesMoscowRussia

Personalised recommendations