Faster Maxflow via Improved Dynamic Spectral Vertex Sparsifiers

Jan van den Brand; Yu Gao; Arun Jambulapati; Yin Tat Lee; Yang P. Liu; Richard Peng; Aaron Sidford

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arXiv · 2112.00722

Faster Maxflow via Improved Dynamic Spectral Vertex Sparsifiers

Jan van den Brand·Yu Gao·Arun Jambulapati·Yin Tat Lee·Yang P. Liu·Richard Peng·Aaron Sidford

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Abstract

We make several advances broadly related to the maintenance of electrical flows in weighted graphs undergoing dynamic resistance updates, including: 1. More efficient dynamic spectral vertex sparsification, achieved by faster length estimation of random walks in weighted graphs using Morris counters [Morris 1978, Nelson-Yu 2020]. 2. A direct reduction from detecting edges with large energy in dynamic electric flows to dynamic spectral vertex sparsifiers. 3. A procedure for turning algorithms for estimating a sequence of vectors under updates from an oblivious adversary to one that tolerates adaptive adversaries via the Gaussian-mechanism from differential privacy. Combining these pieces with modifications to prior robust interior point frameworks gives an algorithm that on graphs with $m$ edges computes a mincost flow with edge costs and capacities in $[1, U]$ in time $\widetilde{O}(m^{3/2-1/58} \log^2 U)$. In prior and independent work, [Axiotis-M\k{a}dry-Vladu FOCS 2021] also obtained an improved algorithm for sparse mincost flows on capacitated graphs. Our algorithm implies a $\widetilde{O}(m^{3/2-1/58} \log U)$ time maxflow algorithm, improving over the $\widetilde{O}(m^{3/2-1/328}\log U)$ time maxflow algorithm of [Gao-Liu-Peng FOCS 2021].

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Jan van den Brand, Yu Gao, Arun Jambulapati, Yin Tat Lee, Yang P. Liu, Richard Peng, Aaron Sidford. 2021-12-01. Faster Maxflow via Improved Dynamic Spectral Vertex Sparsifiers. https://arxiv.org/abs/2112.00722

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