Sublinear time algorithms for approximating maximum matching size have long been studied. Much of the progress over the last two decades on this problem has been on the algorithmic side. For instance, an algorithm of Behnezhad [FOCS'21] obtains a 1/2-approximation in $\tilde{O}(n)$ time for $n$-vertex graphs. A more recent algorithm by Behnezhad, Roghani, Rubinstein, and Saberi [SODA'23] obtains a slightly-better-than-1/2 approximation in $O(n^{1+\epsilon})$ time. On the lower bound side, Parnas and Ron [TCS'07] showed 15 years ago that obtaining any constant approximation of maximum matching size requires $\Omega(n)$ time. Proving any super-linear in $n$ lower bound, even for $(1-\epsilon)$-approximations, has remained elusive since then. In this paper, we prove the first super-linear in $n$ lower bound for this problem. We show that at least $n^{1.2 - o(1)}$ queries in the adjacency list model are needed for obtaining a $(\frac{2}{3} + \Omega(1))$-approximation of maximum matching size. This holds even if the graph is bipartite and is promised to have a matching of size $\Theta(n)$. Our lower bound argument builds on techniques such as correlation decay that to our knowledge have not been used before in proving sublinear time lower bounds. We complement our lower bound by presenting two algorithms that run in strongly sublinear time of $n^{2-\Omega(1)}$. The first algorithm achieves a $(\frac{2}{3}-\epsilon)$-approximation; this significantly improves prior close-to-1/2 approximations. Our second algorithm obtains an even better approximation factor of $(\frac{2}{3}+\Omega(1))$ for bipartite graphs. This breaks the prevalent $2/3$-approximation barrier and importantly shows that our $n^{1.2-o(1)}$ time lower bound for $(\frac{2}{3}+\Omega(1))$-approximations cannot be improved all the way to $n^{2-o(1)}$.

翻译：用于接近最大匹配规模的亚线时间算法 {1.2 长期研究。 在过去20年中, 这个问题上的大部分进展都在算法方面。 例如, Behnezhad [FOCS'21] 的算法在$\ tilde{O} (n) 美元上取得了1/2 的一致。 由Behnezhad、 Roghani、 Rubinstein 和 Saberi [SODA'23] 提供的较新的算法在美元( n=1) 3 美元上略高于-1/2 的近距离; 由Behnez 3 的算法在运算法方面进展相当大; 例如, 由Parnas 和 Ron [TCS'07] 获得最大匹配规模的一致值,需要$(m) 美元(n) 。 以美元( 美元) 预估测任何超线值的算法, 即使是$( i- licl) 美元(l) 和 美元(dro) codeal) 23 的算, 自那时, 我们的第一次的算上, 美元(l=1美元===xxxxxxxx 需要的比现在更需要的显示。