We study the problem of estimating the size of the maximum matching in the sublinear-time setting. This problem has been extensively studied, with several known upper and lower bounds. A notable result by Behnezhad (FOCS 2021) established a 2-approximation in ~O(n) time. However, all known upper and lower bounds are in the adaptive query model, where each query can depend on previous answers. In contrast, non-adaptive query models-where the distribution over all queries must be fixed in advance-are widely studied in property testing, often revealing fundamental gaps between adaptive and non-adaptive complexities. This raises the natural question: is adaptivity also necessary for approximating the maximum matching size in sublinear time? This motivates the goal of achieving a constant or even a polylogarithmic approximation using ~O(n) non-adaptive adjacency list queries, similar to what was done by Behnezhad using adaptive queries. We show that this is not possible by proving that any randomized non-adaptive algorithm achieving an n^{1/3 - gamma}-approximation, for any constant gamma > 0, with probability at least 2/3, must make Omega(n^{1 + eps}) adjacency list queries, for some constant eps > 0 depending on gamma. This result highlights the necessity of adaptivity in achieving strong approximations. However, non-trivial upper bounds are still achievable: we present a simple randomized algorithm that achieves an n^{1/2}-approximation in O(n log^2 n) queries. Moreover, our lower bound also extends to the newly defined variant of the non-adaptive model, where queries are issued according to a fixed query tree, introduced by Azarmehr, Behnezhad, Ghafari, and Sudan (FOCS 2025) in the context of Local Computation Algorithms.

翻译：我们研究了在亚线性时间设置下估计最大匹配大小的问题。该问题已被广泛研究，存在多个已知的上界和下界。Behnezhad（FOCS 2021）的一个显著结果在 ~O(n) 时间内实现了 2-近似。然而，所有已知的上界和下界均属于自适应查询模型，其中每次查询可以依赖于先前答案。相比之下，非自适应查询模型——所有查询的分布必须预先固定——在性质测试中被广泛研究，通常揭示了自适应与非自适应复杂度之间的根本性差距。这自然引出一个问题：在亚线性时间内近似最大匹配大小时，自适应性是否也是必要的？这激发了我们的目标：使用 ~O(n) 次非自适应邻接表查询实现常数甚至多对数近似，类似于 Behnezhad 使用自适应查询所达到的效果。我们证明这是不可能的：对于任意常数 gamma > 0，任何以至少 2/3 的概率实现 n^{1/3 - gamma}-近似的随机化非自适应算法，必须进行 Omega(n^{1 + eps}) 次邻接表查询，其中常数 eps > 0 依赖于 gamma。这一结果凸显了在实现强近似时自适应性的必要性。然而，非平凡的上界仍然可以实现：我们提出了一种简单的随机化算法，在 O(n log^2 n) 次查询内实现 n^{1/2}-近似。此外，我们的下界也适用于新定义的非自适应模型变体，其中查询按照固定查询树发出，该模型由 Azarmehr、Behnezhad、Ghafari 和 Sudan（FOCS 2025）在局部计算算法背景下引入。