Fully Dynamic Matching and Ordered Ruzsa-Szemer\'edi Graphs

TL;DR

This paper introduces Ordered Ruzsa-Szemerédi graphs and links their properties to the complexity of maintaining approximate maximum matchings in fully dynamic graphs, potentially advancing the understanding of optimal update times.

Contribution

It establishes a novel connection between dynamic matching algorithms and ORS graphs, proposing a new randomized algorithm with near-optimal update time based on ORS graph properties.

Findings

Introduces Ordered Ruzsa-Szemerédi graphs as a key concept.

Provides a new randomized algorithm with improved update time.

Links the complexity of dynamic matching to ORS graph bounds.

Abstract

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 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 [FOCS'24]. Whether this can be improved to $n^{1-\Omega(1)}$ remains a major open problem. In this paper, we introduce {\em Ordered Ruzsa-Szemer\'edi (ORS)} graphs (a generalization of Ruzsa-Szemer\'edi graphs) and show that the complexity of dynamic matching is closely tied to them. For $\delta > 0$, define $ORS(\delta n)$ to be the maximum number of matchings $M_1, \ldots, M_t$, each of size $\delta n$, that one can pack in an $n$-vertex graph such that each matching $M_i$ is an {\em induced matching} in subgraph $M_1 \cup \ldots \cup M_{i}$. We show that there is a randomized algorithm that maintains a $(1-\epsilon)$-approximate maximum matching of a fully dynamic graph in $$ \widetilde{O}\left( \sqrt{n^{1+\epsilon} \cdot ORS(\Theta_\epsilon(n))} \right) $$ amortized update-time. While the value of $ORS(\Theta(n))$ remains unknown and is only upper bounded by $n^{1-o(1)}$, the densest construction known from more than two decades ago only achieves $ORS(\Theta(n)) \geq n^{1/\Theta(\log \log n)} = n^{o(1)}$ [Fischer et al. STOC'02]. If this is close to the right bound, then our algorithm achieves an update-time of $\sqrt{n^{1+O(\epsilon)}}$, resolving the aforementioned longstanding open problem in dynamic algorithms in a strong sense.