Maximum st-flow in directed planar graphs via shortest paths

TL;DR

This paper establishes a new duality-based approach linking maximum s-t flows and shortest paths in directed planar graphs without source-sink face constraints, potentially leading to more practical algorithms.

Contribution

It introduces a novel correspondence between maximum flows and shortest paths in directed planar graphs, expanding the applicability beyond previous face-constrained scenarios.

Findings

New duality-based framework for maximum flows in directed planar graphs

Potential for more practical maximum flow algorithms

Extends shortest path relations to general directed planar graphs

Abstract

Minimum cuts have been closely related to shortest paths in planar graphs via planar duality - so long as the graphs are undirected. Even maximum flows are closely related to shortest paths for the same reason - so long as the source and the sink are on a common face. In this paper, we give a correspondence between maximum flows and shortest paths via duality in directed planar graphs with no constraints on the source and sink. We believe this a promising avenue for developing algorithms that are more practical than the current asymptotically best algorithms for maximum st-flow.