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

2026-02-15

AI Summary

This work investigates the feasibility of approximating the size of a maximum matching in a graph under the non-adaptive query model. By leveraging non-adaptive adjacency-list queries, probabilistic analysis, and complexity lower-bound techniques, it establishes—for the first time—that any algorithm achieving an approximation better than \(n^{1/3 - \gamma}\) requires \(\Omega(n^{1+\varepsilon})\) queries, thereby demonstrating the essential role of adaptivity in this problem. Complementing this hardness result, the paper also presents an \(n^{1/2}\)-approximation algorithm that uses only \(O(n \log^2 n)\) queries. These results hold both in the standard non-adaptive query model and in the fixed query tree model, highlighting a sharp separation between adaptive and non-adaptive approaches for graph matching approximation.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

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.

Problem

• sublinear-time
• maximum matching
• non-adaptive queries
• approximation
• lower bounds

Innovation

• non-adaptive queries
• sublinear-time algorithms
• maximum matching approximation
• query complexity lower bounds
• adaptivity gap