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 $(2-\frac{1}{2^{O(1/\gamma)}})$ using $n^{1+\gamma}$ time. This improvement over the factor $2$ is, however, minuscule and they asked if even $1.99$-approximation is possible in $n^{2-\Omega(1)}$ time. We give a strong affirmative answer to this open problem by showing $(1.5+\epsilon)$-approximation algorithms that run in $n^{2-\Theta(\epsilon^{2})}$ 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.

翻译：我们研究估计最大匹配规模的亚线性时间算法。经过一系列长期研究，该问题最终由Behnezhad [FOCS'22]在允许近似因子为$2$的条件下解决。近期，Behnezhad等人[SODA'23]使用$n^{1+\gamma}$时间，将近似因子改进至$(2-\frac{1}{2^{O(1/\gamma)}})$。然而，这一对因子$2$的改进微乎其微，他们提出了一个开放问题：是否可能在$n^{2-\Omega(1)}$时间内实现甚至$1.99$的近似。我们对此问题给出强有力的肯定回答，展示了在$n^{2-\Theta(\epsilon^{2})}$时间内运行的$(1.5+\epsilon)$-近似算法。我们的方法概念上简洁，且与所有先前的亚线性时间匹配算法不同：我们提出一种亚线性时间算法，用于计算边度约束子图（EDCS）的变体。该概念此前已在动态[Bernstein Stein ICALP'15, SODA'16]、分布式[Assadi et al. SODA'19]和流式[Bernstein ICALP'20]设置中被利用，但在亚线性设置中从未出现过。独立工作：Behnezhad、Roghani和Rubinstein [BRR'23]独立展示了与我们定理1.2类似的亚线性算法，涵盖邻接表和矩阵模型。此外，在[BRR'23]中，他们还在上下界两方面展示了关于严格优于$1.5$的近似匹配算法的附加结果。