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A framework for cost-scaling algorithms for submodular flow problems

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1 Author(s)
H. N. Gabow ; Dept. of Comput. Sci., Colorado Univ., Boulder, CO, USA

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{m1/2, n2/3 }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

Published in:

Foundations of Computer Science, 1993. Proceedings., 34th Annual Symposium on

Date of Conference:

3-5 Nov 1993