Deterministic Dynamic Matching in Worst-Case Update Time

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. (Beating the folklore algorithm for dynamic matching, 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 (in: Proceedings of the twenty-seventh annual ACM-SIAM symposium on discrete algorithms, 2016) and Bhattacharya and Kiss (in: 48th international colloquium on automata, languages, and programming, ICALP 2021, 12–16 July, Glasgow, 2021) respectively. The algorithm achieving (3/2 + epsilon ) approximation builds on the EDCS concept introduced by the influential paper of Bernstein and Stein (in: International colloquium on automata, languages, and programming, Springer, Berlin, 2015). Say that H is a (alpha , delta )-approximate matching sparsifier if at all times H satisfies that mu (H) cdot alpha + delta cdot n ge mu (G) (define (alpha , delta )-approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a (3/2 + epsilon , delta )-approximate matching sparsifier. We further show how to reduce the maintenance of an alpha -approximate maximum matching to the maintenance of an (alpha , delta )-approximate maximum matching building based on an observation of Assadi et al. (in: Proceedings of the twenty-seventh annual (ACM-SIAM) symposium on discrete algorithms, (SODA) 2016, Arlington, VA, USA, January 10–12, 2016). Our reduction requires an update time blow-up of {hat{O}}(1) or {tilde{O}}(1) and is deterministic or randomized against an adaptive adversary respectively. To achieve (2 + epsilon )-approximation we improve on the update time guarantee of an algorithm of Bhattacharya and Kiss (in: 48th International colloquium on automata, languages, and programming, ICALP 2021, 12–16 July, Glasgow, 2021). In order to achieve both results we explicitly state a method implicitly used in Nanongkai and Saranurak (in: Proceedings of the twenty-seventh annual ACM symposium on theory of computing, 2017) and Bernstein et al. (Fully-dynamic graph sparsifiers against an adaptive adversary, 2020) which allows to transform dynamic algorithms capable of processing the input in batches to a dynamic algorithms with worst-case update time.