A Faster Deterministic Algorithm for Fully Dynamic Maximal Matching

In the fully dynamic maximal matching problem, the goal is to maintain a maximal matching in a graph undergoing an online sequence of edge insertions and deletions, while minimizing the update time. The problem has been studied extensively in the oblivious-adversary setting, where randomized algorithms with polylogarithmic worst-case and constant amortized update time have been known for some time. A major challenge in this area has been designing an algorithm with nontrivial update time against an adaptive adversary, who may explicitly tailor the update sequence to the algorithm's choices. In a recent breakthrough, Bernstein, Bhattacharya, Kiss, and Saranurak (STOC 2025; hereafter, BBKS25) obtained the first algorithms with sublinear in n update time for this setting: namely, a randomized algorithm with Õ(n 3/4 ) amortized update time, and a deterministic algorithm with Õ(n 8/9 ) amortized update time. Our main result is a deterministic algorithm for fully dynamic maximal matching with amortized update time n 1/2+o (1) .

A powerful tool in dynamic matching is the use of matching sparsifiers: sparse subgraphs that preserve enough information to recover matchings with desired properties. Sparsifiers have been successfully used for approximate maximum matching, yielding sublinear update-time algorithms even against adaptive adversaries. For maximal matching, however, this paradigm is not as natural, since maximality must hold with respect to the entire graph, and so the algorithm must be able to detect and repair violations across all edges. Nevertheless, BBKS25 showed that the EDCS data structure can be ingeniously repurposed as a verification-and-repair mechanism for fully dynamic maximal matching against adaptive adversaries.

We introduce a new deterministic framework, referred to as the subgraph system, which, in contrast to the EDCS data structure used by BBKS25, is purpose-built for verification and maintenance of maximality. The structure of the subgraph system is also carefully designed to allow efficient recursive refinements leading to stronger and stronger parameters. This recursive approach yields our deterministic algorithm with n 1/2+o(1) amortized update time, and provides a new deterministic framework for one of the central graph optimization problems in the dynamic setting.