A framework for cost-scaling algorithms for submodular flow problems

The submodular flow problem includes such problems as minimum-cost network flow, dijoin, edge-connectivity orientation and others. We present a cost-scaling algorithm for submodular flow problems. The algorithm applies to these problems in general; we also examine its efficiency for the dijoin and edge-connectivity orientation problems. A minimum-cost dijoin is found in time O(min{m^1/2, n^2/3 }nmlog(nN)), where n, m and N denote the number of vertices, number of edges and largest magnitude of an integral edge cost. The previous best-known bound is O(n²m) if fast matrix multiplication is not used. A k-edge-connected orientation is found in time O(kn²(√(kn)+k²log(n/k))). A minimum-cost k-edge-connected orientation is found on the above time bound for dijoins when k=O(1) (and a more complicated bound for general k). The scaling algorithm uses a transformation that eliminates vertex weights in edge-capacitated graphs. It also incorporates a scheme to limit the growth in the size of intermediate solutions, using a dual minimum-cost network flow problem