Fast Cycle Canceling Algorithms for Minimum Cost Submodular Flow*

This paper presents two fast cycle canceling algorithms for the submodular flow problem. The first uses an assignment problem whose optimal solution identifies most negative node-disjoint cycles in an auxiliary network. Canceling these cycles lexicographically makes it possible to obtain an optimal submodular flow in O(n 4 h log(nC)) time, which almost matches the current fastest weakly polynomial time for submodular flow (where n is the number of nodes, h is the time for computing an exchange capacity, and C is the maximum absolute value of arc costs). The second algorithm generalizes Goldberg’s cycle canceling algorithm for min cost flow to submodular flow to also get a running time of O(n 4 h log(nC)).. We show how to modify these algorithms to make them strongly polynomial, with running times of O(n 6 h log n), which matches the fastest strongly polynomial time bound for submodular flow. We also show how to extend both algorithms to solve submodular flow with separable convex objectives.

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Correspondence to Satoru Iwata†.

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* An extended abstract of a preliminary version of part of this paper appeared in [22].

† Research supported in part by a Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.

‡ Research supported by an NSERC Operating Grant. Part of this research was done during a sabbatical leave at Cornell SORIE.

§ Research supported in part by a Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.

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Iwata†, S., Mccormick‡, S.T. & Shigeno§, M. Fast Cycle Canceling Algorithms for Minimum Cost Submodular Flow*. Combinatorica 23, 503–525 (2003). https://doi.org/10.1007/s00493-003-0030-3

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AMS Subject Classification (2000):

  • 90C27
  • 90C35
  • 90B10
  • 90C25