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Perfectly matchable subgraph problem on a bipartite graph

Published online by Cambridge University Press:  08 February 2010

Firdovsi Sharifov
Firdovsi Sharifov*
Affiliation:
Glushkov Institute of Cybernetics, Glushkova pr. 40, MSP, 252650 Kyiv, Ukraine; [email protected]
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Abstract

We consider the maximum weight perfectly matchable subgraph problemon a bipartite graph G=(UV,E) with respect to given nonnegativeweights of its edges. We show that G has a perfect matching if andonly if some vector indexed by the nodes in UV is a base of anextended polymatroid associated with a submodular function definedon the subsets of UV. The dual problem of the separation problemfor the extended polymatroid is transformed to the special maximumflow problem on G. In this paper, we give a linear programmingformulation for the maximum weight perfectly matchable subgraphproblem and propose an O(n 3) algorithm to solve it.

Keywords

Bipartite graphextended polymatroidperfect matchingperfectly matchable subgraph
Type
Research Article
Information
RAIRO - Operations Research , Volume 44 , Issue 1 , January 2010 , pp. 27 - 42
Copyright
© EDP Sciences, ROADEF, SMAI, 2010

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References

R. Ahuja, T.K. Magnanti and J.B. Orlin, Network Flows, Theory, Algorithms and Applications. Prentice-Hall (1993).
Alt, H., Blum, N., Mehlhorn, K. and Paul, M., Computing maximum cardinality matching in time $O(n^{1.5}\sqrt{m/\log|V|})$ . Infor. Process. Lett. 37 (1991) 237–240. CrossRef
Balas, E. and Pulleyblank, W., The perfectly matchable subgraph polytope of a bipartite graph. Networks 13 (1983) 495–516. CrossRef
Balas, E. and Pulleyblank, W., The perfectly matchable subgraph polytope of an arbitrary graph. Combinatorica 9 (1989) 321–337. CrossRef
Balinski, M. L., Signature methods for the assignment problem. Oper. Res. 33 (1985) 527–536. CrossRef
Cornaz, D. and Mahjoub, A.R., The Maximum Induced Bipartite Subgraph Problem with Edge Weight. SIAM J. Discrete Math. 3 (2007) 662–675. CrossRef
Cunningham, W.H. and Green-Krotki, J., A separation algorithm for matchable set polytope. Math. Program. 65 (1994) 139–150. CrossRef
M. Grotschel, L. Lovasz and A. Schrijver, Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin (1988).
Sharifov, F.A., Determination of the minimum cut using the base of an extended polymatroid. Cybern. Syst. Anal. 6 (1997) 856–867 (translated from Kibernetika i Systemnyi Analis 6 (1996) 138–152, in Russian).