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 ε n of the optimal solution can be achieved in n2-Ω_ε(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(n²) 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 ε n requires Ω(n2-δ) time in the adjacency list model.
No takes yet. Share an insight, caveat, or question.
Behnezhad et al. (2024) studied this question.