Sublinear-Time Lower Bounds for Approximating Matching Size using Non-Adaptive Queries
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912906531045376 |
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| author | Shah, Vihan |
| author_facet | Shah, Vihan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14326 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sublinear-Time Lower Bounds for Approximating Matching Size using Non-Adaptive Queries Shah, Vihan Data Structures and Algorithms 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. |
| title | Sublinear-Time Lower Bounds for Approximating Matching Size using Non-Adaptive Queries |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2602.14326 |