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(nh 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(nh log(nC)). We show how to modify these algorithms to make them strongly polynomial, with running times of O(nh logn), 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. An extended abstract of a preliminary version of part of this paper appeared in [22]. Department of Mathematical Engineering and Information Physics, University of Tokyo, Tokyo 1138656, Japan ([email protected]). Research supported in part by a Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan. Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, BC V6T 1Z2 Canada. Research supported by an NSERC Operating Grant. Part of this research was done during a sabbatical leave at Cornell SORIE. Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki 305, Japan. Research supported in part by a Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.