An approximate max-flow min-cut relation for undirected multicommodity flow, with applications

Abstract

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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Most of the results in this paper were presented in preliminary form in “Approximation through multicommodity flow”,Proceedings, 31th Annual Symposium on Foundations of Computer Science (1990), pp. 726–737.

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

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Mathematics Subject Classification (1991)

  • 68 Q 25
  • 68 R 10
  • 90 C 08
  • 90 C 27