Flow-cut Dualities for Sheaves on Graphs

This paper generalizes the Max-Flow Min-Cut (MFMC) theorem from the setting of numerical capacities to cellular sheaves of semimodules on directed graphs. Motivating examples of semimodules include probability distributions, multicommodity capacity constraints, and logical propositions. Directed algebraic topology provides the tools necessary for capturing the salient information in such a general setting. First homology classes generalize flows, an orientation sheaf characterizes generalized cuts, first relative homology measures duality gaps, zeroth homology classes generalize both flowvalues and cut-values, and inverse limits generalize infima. Under this dictionary, MFMC is just a special case of a Poincaré Duality for directed topology. A Universal Coefficients Theorem for directed homology generalizes existing criteria for monoid-valued flows to decompose into sums of generalized loops. First homology coincides with a standard generalization of Abelian homology for non-Abelian categories under an assumption of stalkwise flatness, stalkwise module structure, or certain degree bounds on the vertices.