Maximum Flow in Directed Planar Graphs with Vertex Capacities

TL;DR

This paper introduces an efficient O(n log n) algorithm for maximum flow in directed planar graphs with vertex capacities, achieving linear time when source and sink are on the same face, and addresses planarity-preserving reductions.

Contribution

It presents a novel planarity-preserving reduction for maximum flow with vertex capacities and an optimized algorithm for directed planar graphs.

Findings

O(n log n) algorithm for directed planar graphs with vertex capacities

Linear time implementation when source and sink are on the same face

Identification of a flaw in a recent algorithm for undirected planar graphs

Abstract

In this paper we present an O(n log n) algorithm for finding a maximum flow in a directed planar graph, where the vertices are subject to capacity constraints, in addition to the arcs. If the source and the sink are on the same face, then our algorithm can be implemented in O(n) time. For general (not planar) graphs, vertex capacities do not make the problem more difficult, as there is a simple reduction that eliminates vertex capacities. However, this reduction does not preserve the planarity of the graph. The essence of our algorithm is a different reduction that does preserve the planarity, and can be implemented in linear time. For the special case of undirected planar graph, an algorithm with the same time complexity was recently claimed, but we show that it has a flaw.