In this paper, we prove the first approximate max-flow min-cut theorem for undirected multicommodity flow. We show that for a feasible flow to exist in a multicommodity problem, it is sufficient that every cut's capacity exceeds its demand by a factor ofO(logClogD), whereC is the sum of all finite capacities andD is the sum of demands. Moreover, our theorem yields an algorithm for finding a cut that is approximately minimumrelative to the flow that must cross it. We use this result to obtain an approximation algorithm for T. C. Hu's generalization of the multiway-cut problem. This algorithm can in turn be applied to obtain approximation algorithms for minimum deletion of clauses of a 2-CNF≡ formula, via minimization, and other problems. We also generalize the theorem to hypergraph networks; using this generalization, we can handle CNF≡ clauses with an arbitrary number of literals per clause.
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Research supported by the National Science Foundation under NSF grant CDA 8722809, by the Office of Naval and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146, and ARPA Order No. 6320, Amendament 1.
Research supported by NSF grant CCR-9012357 and by an NSF Presidential Young Investigator Award.
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Klein, P., Rao, S., Agrawal, A. et al. An approximate max-flow min-cut relation for undirected multicommodity flow, with applications. Combinatorica 15, 187–202 (1995). https://doi.org/10.1007/BF01200755
Mathematics Subject Classification (1991)
- 68 Q 25
- 68 R 10
- 90 C 08
- 90 C 27