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

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+ root 21 + & varepsilon; approximate to 1.707+& varepsilon; approximation in bipartite graphs and a 1.973+& varepsilon; approximation in general graphs. We thus answer in the affirmative the value version of the major open question repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms' approximation and worst-case update time bounds both hold w.h.p. against adaptive adversaries. 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.