Simple Deterministic Algorithms for Fully Dynamic Maximal Matching

TL;DR

This paper introduces the first deterministic fully dynamic algorithms for maintaining maximal and approximate maximum matchings with improved worst-case update times, advancing the efficiency of dynamic graph algorithms.

Contribution

It presents the first deterministic worst-case update algorithms for fully dynamic maximal and approximate maximum matchings, outperforming naive methods and addressing open problems.

Findings

Deterministic worst-case update time for maximal matching is O(√m).

Maintains a 3/2-approximate maximum matching.

Achieves sub-logarithmic update time for low-arboricity graphs.

Abstract

A maximal matching can be maintained in fully dynamic (supporting both addition and deletion of edges) $n$-vertex graphs using a trivial deterministic algorithm with a worst-case update time of O(n). No deterministic algorithm that outperforms the na\"{\i}ve O(n) one was reported up to this date. The only progress in this direction is due to Ivkovi\'{c} and Lloyd \cite{IL93}, who in 1993 devised a deterministic algorithm with an \emph{amortized} update time of $O((n+m)^{\sqrt{2}/2})$, where $m$ is the number of edges. In this paper we show the first deterministic fully dynamic algorithm that outperforms the trivial one. Specifically, we provide a deterministic \emph{worst-case} update time of $O(\sqrt{m})$. Moreover, our algorithm maintains a matching which is in fact a 3/2-approximate maximum cardinality matching (MCM). We remark that no fully dynamic algorithm for maintaining $(2-\eps)$-approximate MCM improving upon the na\"{\i}ve O(n) was known prior to this work, even allowing amortized time bounds and \emph{randomization}. For low arboricity graphs (e.g., planar graphs and graphs excluding fixed minors), we devise another simple deterministic algorithm with \emph{sub-logarithmic update time}. Specifically, it maintains a fully dynamic maximal matching with amortized update time of $O(\log n/\log \log n)$. This result addresses an open question of Onak and Rubinfeld \cite{OR10}. We also show a deterministic algorithm with optimal space usage, that for arbitrary graphs maintains a maximal matching in amortized $O(\sqrt{m})$ time, and uses only $O(n+m)$ space.