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

TL;DR

This paper introduces a randomized algorithm for fully dynamic graphs that maintains a better-than-2 approximation of maximum matching with near-optimal worst-case update time depending on maximum degree, improving over prior methods.

Contribution

The paper presents the first algorithm achieving a sublinear worst-case update time for maintaining a better-than-2 approximate maximum matching in general dynamic graphs.

Findings

Achieves worst-case update time of O(Δ^ε + polylog n) for any ε > 0.

Improves upon previous algorithms with higher update times such as O(m^{1/4}).

Works with high probability in fully dynamic general graphs.

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].