Beating Greedy Matching in Sublinear Time.

We study sublinear time algorithms for estimating the size of maximum matching in graphs. Our main result is a $(\frac{1}{2}+\Omega(1))$-approximation algorithm which can be implemented in $O(n^{1+\epsilon})$ time, where $n$ is the number of vertices and the constant $\epsilon > 0$ can be made arbitrarily small. The best known lower bound for the problem is $\Omega(n)$, which holds for any constant approximation. Existing algorithms either obtain the greedy bound of $\frac{1}{2}$-approximation [Behnezhad FOCS'21], or require some assumption on the maximum degree to run in $o(n^2)$-time [Yoshida, Yamamoto, and Ito STOC'09]. We improve over these by designing a less "adaptive" augmentation algorithm for maximum matching that might be of independent interest.