2024/06/12 by Soheil Behnezhad, Behnezhad, Soheil, Mohammad Roghani +3 · 2 citations
Computer Science · Mathematics · #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods #Optimization and Search Problems
paper · pdf · doi:10.48550/arxiv.2406.08595
openalex publication_date 2024/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-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(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.