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 є n of the optimal solution can be achieved in n2−Ωє(1) time, where n is the number of vertices. While this is subquadratic in n for any fixed є > 0, it gets closer and closer to the trivial Θ(n2) time algorithm that reads the entire input as є 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 є = є(δ) > 0 such that estimating the size of maximum matching within an additive error of є n requires Ω(n2−δ) time in the adjacency list model.