Deterministic Dynamic Matching In Worst-Case Update Time
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We present deterministic algorithms for maintaining a (3/2 + ϵ) and (2 + ϵ)-approximate maximum matching in a fully dynamic graph with worst-case update times Ô(√(n)) and Õ(1) respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio (2 - δ) (for any δ > 0) and (2 + ϵ) were both shown by Roghani et al. [2021] with update times O(n^3/4) and O_ϵ(√(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_ϵ(√(n)) and Õ(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. Independent Work: Independently and concurrently to our work Grandoni et al. [arXiv'2022] has presented a fully dynamic algorithm for maintaining a (3/2 + ϵ)-approximate maximum matching with deterministic worst-case update time O_ϵ(√(n)).
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