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.
翻译：我们研究亚线性时间算法以估计图表中最大匹配的大小。 我们的主要结果就是 $( frac{ 1 ⁇ 2 ⁇ 2 ⁇ Omega(1)) $- occession 算法, 可以用$( n ⁇ 1 ⁇ 1 ⁇ 2 ⁇ Epsilon}) 美元时间执行, 美元可以任意地使恒定的 $( epsilon) > 0 美元变得小。 最已知的问题下限是 $\ omega( n) 美元, 用于任何恒定的近似值。 现有的算法要么获得$( frac) { 1 ⁇ 2} $( $- accession) 的贪婪绑定 [ Behnezhad FOCS'21], 或要求某种最高程度的假设, 以$( nn ⁇ 2) $( $) y- time[ Yoshida, Yamamoto, Ito STOC'09] 。 我们通过设计一种较不“ 适应性” 增强这些算法来改进这些算法。
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