Approximating Maximum Matching Requires Almost Quadratic Time

TL;DR

This paper proves that estimating maximum matching size within a small additive error in large graphs essentially requires near-quadratic time, confirming the near-optimality of previous subquadratic algorithms.

Contribution

It establishes a near-quadratic lower bound for approximating maximum matching size, closing the gap with prior subquadratic algorithms and showing their near-optimality.

Findings

Estimates within \\varepsilon n require near-quadratic time

Previous subquadratic algorithms are nearly optimal

Lower bounds hold in the adjacency list model

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