Faster Maxflow via Improved Dynamic Spectral Vertex Sparsifiers

TL;DR

This paper introduces improved algorithms for dynamic spectral vertex sparsification and electrical flow maintenance, leading to faster min-cost flow and max-flow computations in weighted graphs with resistance updates.

Contribution

It presents novel methods for dynamic spectral sparsification, energy detection in electric flows, and adversary-tolerant vector estimation, enabling faster flow algorithms.

Findings

Achieves a min-cost flow algorithm with $ ilde{O}(m^{3/2-1/58} \, \log^2 U)$ time.

Derives a max-flow algorithm with $ ilde{O}(m^{3/2-1/58} \, \log U)$ time.

Improves upon previous max-flow algorithms by reducing the exponent in the runtime complexity.

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].