Multiple-source multiple-sink maximum flow in planar graphs

TL;DR

This paper presents an efficient algorithm for computing maximum flow in planar graphs with multiple sources and sinks, achieving a running time that depends only on the number of vertices, which is a significant improvement for such problems.

Contribution

It introduces the first known algorithm with a running time of O(n^(3/2) log^2 n) for this problem in planar graphs, surpassing previous methods.

Findings

Algorithm runs in O(n^(3/2) log^2 n) time for planar graphs.

Multiple-source multiple-sink maximum flow is as hard as the single-source single-sink case in general graphs.

Planarity preservation is crucial for the efficiency of the proposed algorithm.

Abstract

In this paper we show an O(n^(3/2) log^2 n) time algorithm for finding a maximum flow in a planar graph with multiple sources and multiple sinks. This is the fastest algorithm whose running time depends only on the number of vertices in the graph. For general (non-planar) graphs the multiple-source multiple-sink version of the maximum flow problem is as difficult as the standard single-source single-sink version. However, the standard reduction does not preserve the planarity of the graph, and it is not known how to generalize existing maximum flow algorithms for planar graphs to the multiple-source multiple-sink maximum flow problem.