Beating Greedy Matching in Sublinear Time

TL;DR

This paper introduces a sublinear time algorithm that surpasses the greedy approximation ratio for maximum matching in graphs, achieving better results with minimal assumptions and in near-linear time.

Contribution

It presents a novel sublinear time algorithm for maximum matching that improves upon the greedy bound without requiring degree assumptions.

Findings

Achieves a  + \u03b5 approximation in O(n^{1+\u03b5}) time

Surpasses the greedy  approximation barrier in sublinear time

Introduces a less adaptive augmentation algorithm for maximum matching

Abstract

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.