Fully Dynamic Matching: Beating 2-Approximation in ${\Delta }^{\epsilon }$ Update Time

Abstract

In fully dynamic graphs, we know how to maintain a 2-approximation of maximum matching extremely fast, that is, in polylogarithmic update time or better. In a sharp contrast and despite extensive studies, all known algorithms that maintain a $2-\Omega (1)$ approximate matching are much slower. Understanding this gap and, in particular, determining the best possible update time for algorithms providing a better-than-2 approximate matching is a major open question. In this paper, we show that for any constant $\epsilon >0$, there is a randomized algorithm that with high probability maintains a $2-\Omega (1)$ approximate maximum matching of a fully-dynamic general graph in worst-case update time $O({\Delta }^{\epsilon }+\text{polylog }n)$, where $\Delta$ is the maximum degree. Previously, the fastest fully dynamic matching algorithm providing a better-than-2 approximation had $O(m^{1/4})$ update-time [Bernstein and Stein, SODA 2016]. A faster algorithm with update-time $O(n^{\epsilon })$ was known, but worked only for maintaining the size (and not the edges) of the matching in bipartite graphs [Bhattacharya, Henzinger, and Nanongkai, STOC 2016].

A preliminary version of this paper is to appear in proceedings of SODA 2020