The Demand Matching Problem
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.
KeywordsBipartite Graph Knapsack Problem Approximation Guarantee Fractional Edge Node Constraint
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- 1.R. Ahuja, T. Magnanti and J. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice Hall, 1993.Google Scholar
- 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.A. Chakrabarti, C. Chekuri, A. Gupta and A. Kumar, “Approximation algorithms for the unsplittable flow problem”, manuscript, September 2001.Google Scholar
- 4.C. Chekuri and S. Khanna, “A PTAS for the Multiple Knapsack Problem”, Proc. 11th SODA, 2000.Google Scholar
- 5.C. Chekuri, Personal Communication, November 2000.Google Scholar
- 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
- 11.C.H. Papadimitriou, Computational Complexity, Addison Wesley, 1994.Google Scholar
- 13.L. Trevisan, “Non-approximability Results for Optimization Problems on Bounded Degree Instances”, Proc. 33rd STOC, 2001.Google Scholar