Faster Maxflow via Improved Dynamic Spectral Vertex Sparsifiers
(2112.00722)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 $\widetilde{O}(m{3/2-1/58} \log2 U)$. 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].
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.