Maximum Integer Flows in Directed Planar Graphs with Multiple Sources and Sinks and Vertex Capacities

TL;DR

This paper introduces three efficient algorithms for maximum flow problems in directed planar graphs with multiple sources, sinks, and capacities, improving computational times especially for small terminal counts and bounded capacities.

Contribution

The paper presents three novel algorithms for maximum flow in planar graphs with multiple sources, sinks, and capacities, achieving near-linear and polynomial time complexities under various conditions.

Findings

First algorithm solves vertex-disjoint paths in near-linear time for bounded capacities.

Second algorithm handles larger capacities with polynomial time complexity.

Third algorithm efficiently computes flows for three terminals with real capacities.

Abstract

We consider the problem of finding maximum flows in planar graphs with capacities on both vertices and edges and with multiple sources and sinks. We present three algorithms when the capacities are integers. The first algorithm runs in $O(n \log^3 n + kn)$ time when all capacities are bounded, where $n$ is the number of vertices in the graph and $k$ is the number of terminals. This algorithm is the first to solve the vertex-disjoint paths problem in near-linear time when $k$ is bounded but larger than 2. The second algorithm runs in $O(k^2(k^3 + \Delta) n \text{ polylog} (nU))$ time, where $U$ is the largest finite capacity of a single vertex and $\Delta$ is the maximum degree of a vertex. Finally, when $k=3$, we present an algorithm that runs in $O(n \log n)$ time; this algorithm works even when the capacities are arbitrary reals. Our algorithms improve on the fastest previously known algorithms when $k$ and $\Delta$ are small and $U$ is bounded by a polynomial in $n$. Prior to this result, the fastest algorithms ran in $O(n^2 / \log n)$ time for real capacities and $O(n^{3/2} \log n \log U)$ for integer capacities.