Flow-Cut Dualities for Sheaves on Graphs

Abstract

This paper generalizes the Max-Flow Min-Cut (MFMC) theorem from the setting of numerical capacities to sheaves of partial semimodules over semirings on directed graphs. Motivating examples of partial semimodules include probability distributions, multicommodity capacity constraints, and logical propositions. Directed (co)homology theories for such sheaves describes familar constructs on networks. First homology classifies locally decomposable flows, an orientation sheaf over a semiring generalizes directions, connecting maps for homology assign values to flows, connecting maps for cohomology assign values to cuts, and a Poincare Duality describes a decomposition of flows as local flows over cuts. A consequent interpretation of feasible flow-values as a homotopy limit generalizes MFMC for edge weights in certain ordered monoids [Frieze] and hence also classical MFMC. Certain duality gaps are explained as a failure for directed sheaf (co)homology to satisfy a natural generalization of exactness.