Faster Algorithm for Maximum Flow in Directed Planar Graphs with Vertex Capacities

TL;DR

This paper presents a significantly faster algorithm for computing maximum flows in directed planar graphs with both arc and vertex capacities, improving previous methods by optimizing flow computation procedures and leveraging advanced push-relabel techniques.

Contribution

It introduces a novel approach that reduces the number of invocations of existing maximum flow algorithms and proposes two faster methods for flow computation in k-apex graphs.

Findings

Achieves an improved time complexity for maximum flow in planar graphs with vertex capacities.

Develops a new sequential implementation of the parallel highest-distance push-relabel algorithm.

Provides faster algorithms for flow computation in k-apex graphs.

Abstract

We give an $O(k^3 n \log n \min(k,\log^2 n) \log^2(nC))$-time algorithm for computing maximum integer flows in planar graphs with integer arc {\em and vertex} capacities bounded by $C$, and $k$ sources and sinks. This improves by a factor of $\max(k^2,k\log^2 n)$ over the fastest algorithm previously known for this problem [Wang, SODA 2019]. The speedup is obtained by two independent ideas. First we replace an iterative procedure of Wang that uses $O(k)$ invocations of an $O(k^3 n \log^3 n)$-time algorithm for maximum flow algorithm in a planar graph with $k$ apices [Borradaile et al., FOCS 2012, SICOMP 2017], by an alternative procedure that only makes one invocation of the algorithm of Borradaile et al. Second, we show two alternatives for computing flows in the $k$-apex graphs that arise in our modification of Wang's procedure faster than the algorithm of Borradaile et al. In doing so, we introduce and analyze a sequential implementation of the parallel highest-distance push-relabel algorithm of Goldberg and Tarjan~[JACM 1988].