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

TL;DR

It is shown 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.

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.

Figures (12)

• Figure 3: Structure of the observed core subgraph under non-adaptive queries. With high probability, the induced subgraph on the core forms a disjoint union of stars, where edges are directed away from the queried vertex that reveals the edge. Star centers correspond to happy vertices, and petals are not happy. The configurations on the right occur with probability $o(1)$ and are ruled out by \ref{['clm:no-happy-edge', 'clm:no-happy-pairs', 'clm:no-multi-edges']}.
• Figure 4: High-level illustration of the coupling. We progressively fix the randomness in the two distributions in parallel. We omit several steps and only give a flavor of the construction. In particular, we fix the core edges, the labeling permutation, and parts of the adjacency lists. At each stage the supports are restricted until both induce the same observed graph $G_{\text{obs}}$.
• Figure 5: The perfect matching $\mathcal{M} ^*= \mathcal{M} \cup \mathcal{M} '$ pairs instances from $\mathcal{X}$ and $\mathcal{Y}$. For edges in $\mathcal{M}$, the endpoints look identical, so any algorithm succeeds with probability $1/2$. Edges in $\mathcal{M} '$ form only a $o(1)$ fraction, and their endpoints may be distinguished with probability at most $1$. Hence the overall success probability is at most $1/2+o(1)$.
• Figure : (a) Yes Instance
• Figure : (a) Yes Instance ($\mathcal{D}_{\text{\tiny YES}}$)

Theorems & Definitions (92)

• Theorem 1
• Theorem 2
• Corollary 1.1
• Theorem 3
• Theorem 4
• Corollary 1.2
• Corollary 1.3
• Proposition 2.1: Chernoff bound; c.f. dubhashi1996ballsdubhashi2009concentration
• Theorem 4
• Claim 3.2