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 (polylog n) time or a $2/3$-approximate matching in $O(\sqrt{n})$ time, where $n$ is the number of vertices. Improving either of these bounds has been a long-standing open problem. In this paper, we show that when the goal is to maintain just the size of the matching instead of its edge-set, then these bounds can indeed be improved. We give algorithms that maintain * a .501-approximation in (polylog n) update-time for general graphs, * a .534-approximation in (polylog n) update-time for bipartite graphs, and * a $(\frac{2}{3} + \Omega(1))$-approximation in $O(\sqrt{n})$ update-time for bipartite graphs.