The Demand Matching Problem

  • Bruce Shepherd
  • Adrian Vetta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)


We examine formulations for the well-known b-matching problem in the presence of integer demands on the edges. A subset M of edges is feasible if for each node v, the total demand of edges in M incident to it is at most b v. We examine the system of star inequalities for this problem. This system yields an exact linear description for b-matchings in bipartite graphs. For the demand version, we show that the integrality gap for this system is at least 2 1/2 and at most 2 13/16. For general graphs, the gap lies between 3 and 3 5/16. A fully polynomial approximation scheme is also presented for the problem on a tree, thus generalizing a well-known result for the knapsack problem.


Bipartite Graph Knapsack Problem Approximation Guarantee Fractional Edge Node Constraint 
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.
    R. Ahuja, T. Magnanti and J. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice Hall, 1993.Google Scholar
  2. 2.
    P. Berman and M. Karpinksi, “On some tighter inapproximability results”, Proc. 26 th Int. Coll. on Automota, Languages, and Programming, pp200–209, 1999.Google Scholar
  3. 3.
    A. Chakrabarti, C. Chekuri, A. Gupta and A. Kumar, “Approximation algorithms for the unsplittable flow problem”, manuscript, September 2001.Google Scholar
  4. 4.
    C. Chekuri and S. Khanna, “A PTAS for the Multiple Knapsack Problem”, Proc. 11th SODA, 2000.Google Scholar
  5. 5.
    C. Chekuri, Personal Communication, November 2000.Google Scholar
  6. 6.
    Y. Dinitz, N. Garg and M. Goemans, “On the Single-Source Unsplittable Flow Problem”, Combinatorica, 19, pp17–41, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    J. Edmonds, “Maximum matching and a polyhedron with 0, 1-vertices”, Journal of Research of the Natuional Bureau of Standards (B), 69, pp125–30, 1965.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    N. Garg, V.V. Vazirani and M. Yannakakis, “Primal-Dual Approximation Algorithms for Integral Flow and Multicut in Trees with Applications to Matching and Set Cover”, Algorithmica 18, pp3–20, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    A.J. Hoffman and J.B. Kruskal, “Integral boundary points of convex polyhedra”, in H.W. Kuhn and A.W. Tucker eds., Linear Inequalities and Related Systems, Princeton University Press, pp223–246, 1956.Google Scholar
  10. 10.
    O.H. Ibarra and C.E. Kim, “Fast approximation for the knapsack and sum of subset problems”, Journal of the ACM, 22, pp463–468, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    C.H. Papadimitriou, Computational Complexity, Addison Wesley, 1994.Google Scholar
  12. 12.
    D. Shmoys and É. Tardos, “An approximation algorithm for the generalized assignment problem”, Mathematical Programming A, 62, pp461–474, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    L. Trevisan, “Non-approximability Results for Optimization Problems on Bounded Degree Instances”, Proc. 33rd STOC, 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Bruce Shepherd
    • 1
  • Adrian Vetta
    • 2
  1. 1.Bell LaboratoriesUSA
  2. 2.Massachusetts Institute of TechnologyUSA

Personalised recommendations