Fully Dynamic Matching and Ordered Ruzsa-Szemerédi Graphs
Soheil Behnezhad, Alma Ghafari
TL;DR
This paper shows that there is a randomized algorithm that maintains a $(1-\varepsilon)$ -approximate maximum matching of a fully dynamic graph in amortized update-time and introduces Ordered Ruzsa-Szemerédi (ORS) graphs and shows that the complexity of dynamic matching is closely tied to them.
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 $(1-ε)$-approximate maximum matching for an arbitrarily small constant $ε> 0$. Until recently, the fastest known algorithm for this problem required $Θ(n)$ time per update where $n$ is the number of vertices. This bound was slightly improved to $n/(\log^* n)^{Ω(1)}$ by Assadi, Behnezhad, Khanna, and Li [STOC'23] and very recently to $n/2^{Ω(\sqrt{\log n})}$ by Liu [FOCS'24]. Whether this can be improved to $n^{1-Ω(1)}$ remains a major open problem. In this paper, we introduce {\em Ordered Ruzsa-Szemerédi (ORS)} graphs (a generalization of Ruzsa-Szemerédi graphs) and show that the complexity of dynamic matching is closely tied to them. For $δ> 0$, define $ORS(δn)$ to be the maximum number of matchings $M_1, \ldots, M_t$, each of size $δn$, 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 $(1-ε)$-approximate maximum matching of a fully dynamic graph in $$ \widetilde{O}\left( \sqrt{n^{1+ε} \cdot ORS(Θ_ε(n))} \right) $$ amortized update-time. While the value of $ORS(Θ(n))$ remains unknown and is only upper bounded by $n^{1-o(1)}$, the densest construction known from more than two decades ago only achieves $ORS(Θ(n)) \geq n^{1/Θ(\log \log n)} = n^{o(1)}$ [Fischer et al. STOC'02]. If this is close to the right bound, then our algorithm achieves an update-time of $\sqrt{n^{1+O(ε)}}$, resolving the aforementioned longstanding open problem in dynamic algorithms in a strong sense.