Deterministic Dynamic Matching In Worst-Case Update Time

TL;DR

This paper introduces deterministic algorithms for maintaining approximate maximum matchings in dynamic graphs with improved worst-case update times, closing the gap with existing amortized algorithms and explicitly detailing a batch-to-worst-case update transformation.

Contribution

It presents new deterministic algorithms with worst-case update times for approximate maximum matching, bridging the gap with prior amortized algorithms and clarifying a batch processing method.

Findings

Achieved deterministic worst-case update times of $ ilde{O}( oot{2}n)$ and $ ilde{O}(1)$ for different approximation ratios.

Closed the gap between worst-case and amortized update times for maximum matching.

Provided a method to convert batch-processing algorithms into worst-case update algorithms.

Abstract

We present deterministic algorithms for maintaining a $(3/2 + \epsilon)$ and $(2 + \epsilon)$-approximate maximum matching in a fully dynamic graph with worst-case update times $\hat{O}(\sqrt{n})$ and $\tilde{O}(1)$ respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio $(2 - \delta)$ (for any $\delta > 0$) and $(2 + \epsilon)$ were both shown by Roghani et al. [2021] with update times $O(n^{3/4})$ and $O_\epsilon(\sqrt{n})$ respectively. We close the gap between worst-case and amortized algorithms for the two approximation ratios as the best deterministic amortized update times for the problem are $O_\epsilon(\sqrt{n})$ and $\tilde{O}(1)$ which were shown in Bernstein and Stein [SODA'2021] and Bhattacharya and Kiss [ICALP'2021] respectively. In order to achieve both results we explicitly state a method implicitly used in Nanongkai and Saranurak [STOC'2017] and Bernstein et al. [arXiv'2020] which allows to transform dynamic algorithms capable of processing the input in batches to a dynamic algorithms with worst-case update time. \textbf{Independent Work:} Independently and concurrently to our work Grandoni et al. [arXiv'2021] has presented a fully dynamic algorithm for maintaining a $(3/2 + \epsilon)$-approximate maximum matching with deterministic worst-case update time $O_\epsilon(\sqrt{n})$.