On the Effectiveness of the Linear Programming Relaxation of the 0-1 Multi-commodity Minimum Cost Network Flow Problem

  • Dae-Sik Choi
  • In-Chan Choi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4112)


Several studies have reported that the linear program relaxation of integer multi-commodity network flow problems often provides integer optimal solutions. We explore this phenomenon with a 0-1 multi-commodity network with mutual arc capacity constraints. Characteristics of basic solutions in the linear programming relaxation problem of the 0-1 multi-commodity problem are identified. Specifically, necessary conditions for a linear programming relaxation to have a non-integer solution are presented. Based on the observed characteristics, a simple illustrative example problem is constructed to show that its LP relaxation problem has integer optimal solutions with a relatively high probability. Furthermore, to investigate whether or not and under what conditions this tendency applies to large-sized problems, we have carried out computational experiments by using randomly generated problem instances. The results of our computational experiment indicate that there exists a narrow band of arc density in which the 0-1 multi-commodity problems possess no integer optimal solutions.


Linear Program Relaxation Basic Feasible Solution Vehicle Schedule Integer Optimal Solution Multicommodity Network 
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.
    Brandes, U., Schlickenrieder, W., Neyer, G., Wagner, D., Weihe, K.: PlaNet, a Software Package of Algorithms and Heuristics for Disjoint Paths in Planar Networks. Discrete Applied Mathematics 92, 91–110 (1999)MATHCrossRefGoogle Scholar
  2. 2.
    Cappanera, P., Gallo, G.: A Multicommodity Flow Approach to the Crew Rostering Problem. Operations Research 52, 583–596 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Desauliers, G., Desrosiers, J., Dumas, Y., Solomon, M.M., Soumis, F.: Daily Aircraft Routing and Scheduling. Management Science 43, 841–855 (1997)CrossRefGoogle Scholar
  4. 4.
    Evans, J.R.: A Single Commodity Transformation for Certain Multicommodity Networks. Operations Research 26, 673–680 (1978)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Faneyte, D.B.C., Spieksma, F.C.R., Woeginger, G.J.: A Branch-and-price Algorithm for a Hierarchical Crew Scheduling Problems. Naval Research Logistics 49, 743–759 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Farvoleden, J.M., Powell, W.B., Lustig, I.J.: A primal Solution for the Arc-chain Formulation of a Multicommodity Network Flow Problem. Operations Research 41, 669–693 (1993)CrossRefGoogle Scholar
  7. 7.
    Gouveia, L.: Multicommodity Flow Models for Spanning Trees with Hop Constraints. European Journal of Operational Research 95, 178–190 (1996)MATHCrossRefGoogle Scholar
  8. 8.
    Kleitmann, D.J., Martin-Lof, A., Rothschild, B., Whinston, A.: A Matching Theorem for Graphs. Journal of Combinatorial Theory 8, 104–114 (1970)CrossRefGoogle Scholar
  9. 9.
    Löbel, A.: Vehicle Scheduling in Public Transit and Lagrangean Pricing. Management Science 44, 1637–1649 (1998)MATHCrossRefGoogle Scholar
  10. 10.
    Maurras, J.-F., Vaxès, Y.: Multicommodity Network Flow with Jump Constraints. Discrete Mathematics 165/166, 481–486 (1997)CrossRefGoogle Scholar
  11. 11.
    Moz, M., Pato, M.V.: An Integer Multicommodity Flow Model Applied to the Rerostering of Nurse Schedules. Annals of Operations Research 119, 285–301 (2003)MATHCrossRefGoogle Scholar
  12. 12.
    Ozdaglar, A.E., Bertsekas, D.P.: Optimal Solution of Integer Multicommodity Flow Problems with Application in Optical Networks. In: Proc. of Symposium on Global Optimization, Santorini, Greece (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Dae-Sik Choi
    • 1
  • In-Chan Choi
    • 1
  1. 1.Department of Industrial Systems and Information EngineeringKorea UniversitySeoulSouth Korea

Personalised recommendations