A Polynomial Algorithm for Weighted Abstract Flow

Ford and Fulkerson’s original 1956 max flow/min cut paper formulated max flow in terms of flows on paths, rather than the more familiar flows on arcs. In 1974 Hoffman pointed out that Ford and Fulkerson’s original proof was quite abstract, and applied to a wide range of flow problems. In this abstract model we have capacitated elements, and linearly ordered subsets of elements called paths. When two paths share an element (“cross”), then there must be a path that is a subset of the first path up to the cross and the second path after the cross. Hoffman’s generalization of Ford and Fulkerson’s proof showed that integral optimal primal and dual solutions still exist under this weak assumption. However, his proof is non-constructive. Hoffman’s paper considers a sort of supermodular objective on the path flows, which allows him to include transportation problems and thus min-cost flow in his framework. We develop the first combinatorial polynomial algorithm that solves this problem, thereby also give a constructive proof of Hoffman’s theorem. Our algorithm accesses the network only through a dual feasibility oracle, and resembles the successive shortest path algorithm for ordinary min-cost flow. It uses some of the same techniques used to solve the max flow/min cut version of Hoffman’s model, plus a method to re-optimize when capacities change inside capacity scaling.