Approximating Maximum Matching Requires Almost Quadratic Time
<|reference_start|>Approximating Maximum Matching Requires Almost Quadratic Time: We study algorithms for estimating the size of maximum matching. This problem has been subject to extensive research. For $n$-vertex graphs, Bhattacharya, Kiss, and Saranurak [FOCS'23] (BKS) showed that an estimate that is within $\varepsilon n$ of the optimal solution can be achieved in $n^{2-\Omega_\varepsilon(1)}$ time, where $n$ is the number of vertices. While this is subquadratic in $n$ for any fixed $\varepsilon > 0$, it gets closer and closer to the trivial $\Theta(n^2)$ time algorithm that reads the entire input as $\varepsilon$ is made smaller and smaller.   In this work, we close this gap and show that the algorithm of BKS is close to optimal. In particular, we prove that for any fixed $\delta > 0$, there is another fixed $\varepsilon = \varepsilon(\delta) > 0$ such that estimating the size of maximum matching within an additive error of $\varepsilon n$ requires $\Omega(n^{2-\delta})$ time in the adjacency list model.<|reference_end|>
@article{behnezhad2024approximating,
    title={Approximating Maximum Matching Requires Almost Quadratic Time},
    author={Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein},
    journal={arXiv preprint arXiv:2406.08595},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.08595},
    primaryClass={cs.DS}
}