Dynamic Matching with Better-than-2 Approximation in Polylogarithmic Update Time

TL;DR

This paper introduces dynamic algorithms that achieve better-than-2 approximation ratios for maximum matching with polylogarithmic update times, answering longstanding open questions in dynamic graph algorithms.

Contribution

It presents the first dynamic algorithms with approximation ratios below 2 and polylogarithmic update time, using novel simulation of streaming algorithms and sublinear-time matching techniques.

Findings

Achieves 1.707+ε approximation in bipartite graphs.

Achieves 1.973+ε approximation in general graphs.

Algorithms work against adaptive adversaries with worst-case guarantees.

Abstract

We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than $2$. Specifically, we obtain a $1+\frac{1}{\sqrt{2}}+\epsilon\approx 1.707+\epsilon$ approximation in bipartite graphs and a $1.973+\epsilon$ approximation in general graphs. We thus answer in the affirmative the major open question first posed in the influential work of Onak and Rubinfeld (STOC'10) and repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms also work against an adaptive adversary and guarantee worst-case polylog update time, both w.h.p. Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS'21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.