Sublinear-Time Lower Bounds for Approximating Matching Size using Non-Adaptive Queries

TL;DR

This paper investigates the limitations of non-adaptive algorithms in approximating maximum matching size in sublinear time, establishing lower bounds that demonstrate adaptivity's necessity for strong approximations, while also providing a simple non-adaptive approximation algorithm.

Contribution

It proves that non-adaptive algorithms cannot achieve better than roughly n^{1/3} approximation with sublinear queries, highlighting the importance of adaptivity, and presents a non-adaptive algorithm with an n^{1/2} approximation.

Findings

Non-adaptive algorithms require Omega(n^{1+eps}) queries for n^{1/3 - gamma}-approximation.

A simple non-adaptive algorithm achieves an n^{1/2}-approximation with O(n log^2 n) queries.

Lower bounds extend to fixed-query-tree models in local computation algorithms.

Abstract

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.