Max-Flow on Regular Spaces

TL;DR

This paper extends the max-flow problem to regular spaces, revealing a submodular structure and generalizing classical algorithms with quadratic bounds on augmentations.

Contribution

It introduces a generalized max-flow framework on regular spaces and adapts the Edmonds-Karp algorithm with improved augmentation bounds.

Findings

Submodular structure of path families in regular spaces

Generalization of Edmonds-Karp algorithm

Quadratic bound on the number of augmentations

Abstract

The max-flow and max-coflow problem on directed graphs is studied in the common generalization to regular spaces, i.e., to kernels or row spaces of totally unimodular matrices. Exhibiting a submodular structure of the family of paths within this model we generalize the Edmonds-Karp variant of the classical Ford-Fulkerson method and show that the number of augmentations is quadratically bounded if augmentations are chosen along shortest possible augmenting paths.