Fully Dynamic Matching and Ordered Ruzsa-Szemerédi Graphs - Behnezhad, Soheil; Ghafari, Alma (Proceedings / annual Symposium on Foundations of Computer Science, 2024)
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-\varepsilon) -approximate maximum matching for an arbitrarily small constant \varepsilon > 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^{\ast}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 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 \delta > 0 , define ORS (\delta n) to be the maximum number of matchings M_{1}, \ldots, 1M_{t} , each of size \delta n , that one can pack in an n-vertex graph such that each matching M_{i} is an induced matching in subgraph M_{1}\cup\ldots\cup M_{i} . We show that there is a randomized algorithm that maintains a (1-\varepsilon) -approximate maximum matching of a fully dynamic graph in amortized update-time. While the value of \text{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(\varepsilon)}}^{-} , resolving the aforementioned longstanding open problem in dynamic algorithms in a strong sense.