Beating Greedy Matching in Sublinear Time
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We study sublinear time algorithms for estimating the size of maximum matching in graphs. Our main result is a (1/2+Ω(1))-approximation algorithm which can be implemented in O(n^1+ϵ) time, where n is the number of vertices and the constant ϵ > 0 can be made arbitrarily small. The best known lower bound for the problem is Ω(n), which holds for any constant approximation. Existing algorithms either obtain the greedy bound of 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.
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