Abstract

We present a simplex method for the solution of the optimal submodular flow problem. Like the network simplex method for solving the minimum cost network flow problem, this algorithm is purely combinatorial. It requires an oracle which can minimize a submodular function. In general this oracle is available only via the ellipsoid algorithm, but in several applications it is provided by an efficient combinatorial procedure. We generalize the notion of strong feasibity in the network simplex method to give a finiteness proof for the new algorithm. Hence we obtain a constructive proof of the integrality theorem of Edmonds and Giles.

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