A faster capacity scaling algorithm for minimum cost submodular flow

We describe an O(nhmin{logU, n log n}) capacity scaling algorithm for the minimum cost submodular flow problem. Our algorithm modifies and extends the Edmonds–Karp capacity scaling algorithm for minimum cost flow to solve the minimum cost submodular flow problem. The modification entails scaling a relaxation parameter δ. Capacities are relaxed by attaching a complete directed graph with uniform arc capacity δ in each scaling phase. We then modify a feasible submodular flow by relaxing the submodular constraints, so that complementary slackness is satisfied. This creates discrepancies between the boundary of the flow and the base polyhedron of a relaxed submodular function. To reduce these discrepancies, we use a variant of the successive shortest path algorithm that augments flow along minimum cost paths of residual capacity at least δ. The shortest augmenting path subroutine we use is a variant of Dijkstra’s algorithm modified to handle exchange capacity arcs efficiently. The result is a weakly polynomial time algorithm whose running time is better than any existing submodular flow algorithm when U is small and C is big. We also show how to use maximum mean cuts to make the algorithm strongly polynomial. The resulting algorithm is the first capacity scaling algorithm to match the current best strongly polynomial bound for submodular flow. ∗Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Email: [email protected]. This research was done in part while visiting RIMS, Kyoto University, with support by Grant-in-Aid for Scientific Research No. 10750053 from Ministry of Education, Science, Sports, and Culture of Japan, and while on leave at CORE, Université catholique de Louvain, Belgium. Additional support from NSF through grant EIA-0049084. †Department of Mathematical Engineering and Information Physics, University of Tokyo, Tokyo 113-8656, Japan. Email: [email protected]. Partially supported by Grant-in-Aid for Scientific Research No. 10750053 from Ministry of Education, Science, Sports, and Culture of Japan. ‡Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, BC V6T 1Z2 Canada. Email: [email protected]. Research supported by an NSERC Operating Grant; part of this research was done during visits to SORIE at Cornell University and to LIMOS at Université Blaise Pascal, Clermont-Ferrand.