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 εn of the optimal solution can be achieved in n 2-Ωε(1) time, where n is the number of vertices. While this is subquadratic in n for any fixed ε > 0, it gets closer and closer to the trivial Θ(n 2 ) time algorithm that reads the entire input as ε 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 δ > 0, there is another fixed ε = ε(δ) > 0 such that estimating the size of maximum matching within an additive error of εn requires Ω(n 2-δ ) time in the adjacency list model.