Sublinear Algorithms for (1.5+ε)-Approximate Matching
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We study sublinear time algorithms for estimating the size of maximum matching. After a long line of research, the problem was finally settled by Behnezhad [FOCS'22], in the regime where one is willing to pay an approximation factor of 2. Very recently, Behnezhad et al.[SODA'23] improved the approximation factor to (2-1/2^O(1/γ)) using n^1+γ time. This improvement over the factor 2 is, however, minuscule and they asked if even 1.99-approximation is possible in n^2-Ω(1) time. We give a strong affirmative answer to this open problem by showing (1.5+ϵ)-approximation algorithms that run in n^2-Θ(ϵ^2) time. Our approach is conceptually simple and diverges from all previous sublinear-time matching algorithms: we show a sublinear time algorithm for computing a variant of the edge-degree constrained subgraph (EDCS), a concept that has previously been exploited in dynamic [Bernstein Stein ICALP'15, SODA'16], distributed [Assadi et al. SODA'19] and streaming [Bernstein ICALP'20] settings, but never before in the sublinear setting. Independent work: Behnezhad, Roghani and Rubinstein [BRR'23] independently showed sublinear algorithms similar to our Theorem 1.2 in both adjacency list and matrix models. Furthermore, in [BRR'23], they show additional results on strictly better-than-1.5 approximate matching algorithms in both upper and lower bound sides.
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