Dynamic Algorithms for Maximum Matching Size

We study fully dynamic algorithms for maximum matching. This is a well-studied problem, known to admit several update-time/approximation trade-offs. For instance, it is known how to maintain a 1/2-approximate matching in $\log^{O(1)} n$ update time or a $2/3$-approximate matching in $O(\sqrt{n})$ update time, where $n$ is the number of vertices. It has been a long-standing open problem to determine whether either of these bounds can be improved. In this paper, we show that when the goal is to maintain just the size of the matching (and not its edge-set), then these bounds can indeed be improved. First, we give an algorithm that takes $\log^{O(1)} n$ update-time and maintains a $.501$-approximation ($.585$-approximation if the graph is bipartite). Second, we give an algorithm that maintains a $(2/3 + Ω(1))$-approximation in $O(\sqrt{n})$ time for bipartite graphs. Our results build on new connections to sublinear time algorithms. In particular, a key tool for both is an algorithm of the author for estimating the size of maximal matchings in $\widetilde{O}(n)$ time [Behnezhad; FOCS 2021]. Our second result also builds on the edge-degree constrained subgraph (EDCS) of Bernstein and Stein [ICALP'15, SODA'16]. In particular, while it has been known that EDCS may not include a better than 2/3-approximation, we give a new characterization of such tight instances which allows us to break it. We believe this characterization might be of independent interest.