Approximating Maximum Matching Requires Almost Quadratic Time

Abstract: 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 showed that an estimate that is within $\varepsilon n$ of the optimal solution can be achieved in $n<sup>{2-\Omega_\varepsilon(1)}$ time, where $n$ is the number of vertices. While this is subquadratic in $n$ for any fixed $\varepsilon &gt; 0$, it gets closer and closer to the trivial $\Theta(n<sup>2)$ 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 $\delta &gt; 0$, there is another fixed $\varepsilon = \varepsilon(\delta) &gt; 0$ such that estimating the size of maximum matching within an additive error of $\varepsilon n$ requires $\Omega(n<sup>{2-\delta})$ time in the adjacency list model.