Fully Dynamic Maximal Matching in Constant Update Time

TL;DR

This paper introduces a randomized algorithm that maintains a maximal matching in fully dynamic graphs with constant amortized update time, resolving a long-standing open problem and improving upon previous logarithmic bounds.

Contribution

The authors present a novel, simpler randomized algorithm achieving constant amortized update time for dynamic maximal matching, improving the theoretical runtime bounds.

Findings

Achieves constant amortized update time for dynamic maximal matching

Maintains 2-approximate vertex cover with constant update time

Uses linear space, improving upon previous algorithms' space complexity

Abstract

Baswana, Gupta and Sen [FOCS'11] showed that fully dynamic maximal matching can be maintained in general graphs with logarithmic amortized update time. More specifically, starting from an empty graph on $n$ fixed vertices, they devised a randomized algorithm for maintaining maximal matching over any sequence of $t$ edge insertions and deletions with a total runtime of $O(t \log n)$ in expectation and $O(t \log n + n \log^2 n)$ with high probability. Whether or not this runtime bound can be improved towards $O(t)$ has remained an important open problem. Despite significant research efforts, this question has resisted numerous attempts at resolution even for basic graph families such as forests. In this paper, we resolve the question in the affirmative, by presenting a randomized algorithm for maintaining maximal matching in general graphs with \emph{constant} amortized update time. The optimal runtime bound $O(t)$ of our algorithm holds both in expectation and with high probability. As an immediate corollary, we can maintain 2-approximate vertex cover with constant amortized update time. This result is essentially the best one can hope for (under the unique games conjecture) in the context of dynamic approximate vertex cover, culminating a long line of research. Our algorithm builds on Baswana et al.'s algorithm, but is inherently different and arguably simpler. As an implication of our simplified approach, the space usage of our algorithm is linear in the (dynamic) graph size, while the space usage of Baswana et al.'s algorithm is always at least $\Omega(n \log n)$. Finally, we present applications to approximate weighted matchings and to distributed networks.