Deterministic dynamic matching in worst-case update time

We present deterministic algorithms for maintaining a (3/2 + ��) and (2 + ��)-approximate maximum matching in a fully dynamic graph with worst-case update times O��(���n) and O��(1) respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio (2 - ��) (for any �� > 0) and (2 + ��) were both shown by Roghani et al. [arXiv'2021] with update times O(n^{3/4}) and O_��(���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_��(���n) and O��(1) which were shown in Bernstein and Stein [SODA'2021] and Bhattacharya and Kiss [ICALP'2021] respectively. The algorithm achieving (3/2 + ��) approximation builds on the EDCS concept introduced by the influential paper of Bernstein and Stein [ICALP'2015]. Say that H is a (��, ��)-approximate matching sparsifier if at all times H satisfies that ��(H) ��� �� + �� ��� n ��� ��(G) (define (��, ��)-approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a (3/2 + ��, ��)-approximate matching sparsifier. We further show how to reduce the maintenance of an ��-approximate maximum matching to the maintenance of an (��, ��)-approximate maximum matching building based on an observation of Assadi et al. [EC'2016]. Our reduction requires an update time blow-up of O��(1) or O��(1) and is deterministic or randomized against an adaptive adversary respectively. To achieve (2 + ��)-approximation we improve on the update time guarantee of an algorithm of Bhattacharya and Kiss [ICALP'2021]. 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. Independent Work: Independently and concurrently to our work Grandoni et al. [arXiv'2021] has presented a fully dynamic algorithm for maintaining a (3/2 + ��)-approximate maximum matching with deterministic worst-case update time O_��(���n).
LIPIcs, Vol. 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), pages 94:1-94:21