A faster algorithm for maximum flow in directed planar graphs with vertex capacities

Joint work with Julian Enoch, Dor Mesica, Shay Mozes

Proceedings of the 32nd International Symposium on Algorithms and Computation (ISAAC), 72:1–72:16, 2021

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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 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].