A Push/Relabel framework for submodular flows and its definement for 0-1 submodular flows
Satoru Fujishige, Xinguang Zhang · Optimization · 1996
We consider the submodular flow problem of Edmonds and Giles. A submodular flow is a flow in a network satisfying capacity constraints and flow-boundary constraints given in terms of the base polyhedron of a submodular system. A cost scaling framework is constructed by using e-optimality concept associated with dual variables of a flow, originally due to Tardos and Bertsekas. The framework is a generalization of Goldberg and Tarjan's push/relabel algorithm for minimum-cost flows and also a generalization of Fujishige and Zhang's algorithm for the submodular intersection problem. Each phase of the cost scaling, called procedure Refine, improves a 2∊-optimal submodular flow to an ∊,-optimal submodular flow. Furthermore, we devise a faster hybrid algorithm of procedure Refine for the 0-1 submodular flow problem which is a natural generalization of Fujishige and Zhang's algorithm for the independent assignment problem. For a network with n vertices, m arcs and integer arc costs bounded by Г, an optimal 0-1 submodular flow can be found in time by our algorithm under oracles for the dependence function and the exchange capacity of the given submodular system