Deterministic Dynamic Maximal Matching in Sublinear Update Time

2025-04-29

AI Summary

This work breaks the long-standing deterministic lower bound on update time for dynamic maximum matching in dense graphs. Previously, all deterministic algorithms required Ω(n) amortized update time on n-vertex dense graphs. We present the first deterministic dynamic algorithm that maintains a maximal matching in Õ(n⁸⁄₉) amortized time—surpassing the linear barrier. Our approach introduces three key techniques: (1) repurposing the Edge Degree Constrained Subgraph (EDCS) to guarantee full matching of high-degree vertices, diverging from its conventional use for approximation; (2) integrating sublinear-time matching computation, random walks on directed expander graphs, and monotonic Even–Shiloach trees; and (3) designing a randomized algorithm achieving Õ(n³⁄₄) amortized time against an adaptive adversary. These advances collectively establish new state-of-the-art bounds for deterministic and randomized dynamic maximal matching in dense graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We give a fully dynamic deterministic algorithm for maintaining a maximal matching of an $n$-vertex graph in $ ilde{O}(n^{8/9})$ amortized update time. This breaks the long-standing $Omega(n)$-update-time barrier on dense graphs, achievable by trivially scanning all incident vertices of the updated edge, and affirmatively answers a major open question repeatedly asked in the literature [BGS15, BCHN18, Sol22]. We also present a faster randomized algorithm against an adaptive adversary with $ ilde{O}(n^{3/4})$ amortized update time. Our approach employs the edge degree constrained subgraph (EDCS), a central object for optimizing approximation ratio, in a completely novel way; we instead use it for maintaining a matching that matches all high degree vertices in sublinear update time so that it remains to handle low degree vertices rather straightforwardly. To optimize this approach, we employ tools never used in the dynamic matching literature prior to our work, including sublinear-time algorithms for matching high degree vertices, random walks on directed expanders, and the monotone Even-Shiloach tree for dynamic shortest paths.

Problem

• Deterministic dynamic maximal matching in sublinear update time
• Breaking the Ω(n) update-time barrier on dense graphs
• Using EDCS for matching high degree vertices sublinearly

Innovation

• Uses EDCS for sublinear update time
• Employs sublinear-time high degree matching
• Applies dynamic shortest paths techniques