A Faster Deterministic Algorithm for Fully Dynamic Maximal Matching
2026-05-01 • Data Structures and Algorithms
Data Structures and Algorithms
AI summaryⓘ
The authors study how to keep a maximal matching in a graph that changes over time with edges added or removed. Previous work by Bernstein et al. (BBKS25) gave algorithms that work well even when the changes adapt based on the algorithm’s behavior, but their update times were relatively high. The authors present a new deterministic method that updates faster, with time roughly proportional to the square root of the number of nodes. They introduce a new tool called the subgraph system to efficiently check and fix the matching, improving over earlier tools like EDCS. This leads to a more efficient way to maintain maximal matchings under fully dynamic conditions.
fully dynamic maximal matchingadaptive adversaryamortized update timedeterministic algorithmmatching sparsifiersEDCSsubgraph systemgraph algorithmsrecursive refinementonline edge updates
Authors
Julia Chuzhoy, Sanjeev Khanna, Junkai Song
Abstract
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. 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 non-trivial update time against an adaptive adversary. In a recent breakthrough, Bernstein, Bhattacharya, Kiss, and Saranurak (STOC 2025; hereafter, BBKS25) obtained the first algorithms with sublinear update time for this setting: namely, a randomized algorithm with $\tilde{O}(n^{3/4})$ amortized update time, and a deterministic algorithm with $\tilde{O}(n^{8/9})$ 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, such as the EDCS data structure, have been successfully used for approximate maximum matching. For maximal matching, however, this paradigm is not as natural, since maximality must hold with respect to the entire graph. Nevertheless, BBKS25 showed that EDCS can be 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 EDCS, is purpose-built for verification and maintenance of maximality. It is also designed to allow efficient recursive refinements leading to stronger and stronger parameters, that yield our deterministic algorithm with $n^{1/2+o(1)}$ amortized update time.