We study the fully dynamic maximum matching problem. In this problem, the goal is to efficiently maintain an approximate maximum matching of a graph that is subject to edge insertions and deletions. Our focus is particularly on algorithms that maintain the edges of a $(1-\epsilon)$-approximate maximum matching for an arbitrarily small constant $\epsilon > 0$. Until recently, the fastest known algorithm for this problem required $\Theta(n)$ time per update where $n$ is the number of vertices. This bound was slightly improved to $n/(\log^* n)^{\Omega(1)}$ by Assadi, Behnezhad, Khanna, and Li [STOC'23] and very recently to $n/2^{\Omega(\sqrt{\log n})}$ by Liu [ArXiv'24]. Whether this can be improved to $n^{1-\Omega(1)}$ remains a major open problem. In this paper, we present a new algorithm that maintains a $(1-\epsilon)$-approximate maximum matching. The update-time of our algorithm is parametrized based on the density of a certain class of graphs that we call Ordered Ruzsa-Szemer\'edi (ORS) graphs, a generalization of the well-known Ruzsa-Szemer\'edi graphs. While determining the density of ORS (or RS) remains a hard problem in combinatorics, we prove that if the existing constructions of ORS graphs are optimal, then our algorithm runs in $n^{1/2+O(\epsilon)}$ time for any fixed $\epsilon > 0$ which would be significantly faster than existing near-linear in $n$ time algorithms. Our second main contribution is a better upper bound on density of both ORS and RS graphs with linear size matchings. The previous best upper bound was due to a proof of the triangle-removal lemma from more than a decade ago due to Fox [Annals of Mathematics '11].

翻译：本文研究全动态最大匹配问题。在该问题中，目标是高效维护一个经历边插入和删除的图的最大近似匹配。我们特别关注对于任意小常数$\epsilon > 0$，维护$(1-\epsilon)$-近似最大匹配的边的算法。直到最近，该问题最快的已知算法每次更新需要$\Theta(n)$时间，其中$n$是顶点数。Assadi、Behnezhad、Khanna和Li [STOC'23] 将该界限略微改进至$n/(\log^* n)^{\Omega(1)}$，而Liu [ArXiv'24] 最近进一步改进至$n/2^{\Omega(\sqrt{\log n})}$。能否将其改进至$n^{1-\Omega(1)}$仍是一个重大开放问题。在本文中，我们提出一种新算法，可维护$(1-\epsilon)$-近似最大匹配。该算法的更新时间基于一类称为有序Ruzsa-Szemerédi（ORS）图（Ruzsa-Szemerédi图的一种推广）的图的密度进行参数化。虽然确定ORS（或RS）图的密度仍是组合数学中的一个难题，但我们证明：若现有ORS图构造最优，则对于任何固定$\epsilon > 0$，我们的算法运行时间为$n^{1/2+O(\epsilon)}$，这将显著快于现有的近线性$n$时间算法。我们的第二个主要贡献是对具有线性大小匹配的ORS和RS图密度给出了更好的上界。此前的最佳上界源于十多年前Fox [Annals of Mathematics '11] 对三角形移除引理的证明。