Deterministic Dynamic Matching in Worst-Case Update Time

Abstract

AbstractWe present deterministic algorithms for maintaining a $$(3/2 + \epsilon )$$ ( 3 / 2 + ϵ ) and $$(2 + \epsilon )$$ ( 2 + ϵ ) -approximate maximum matching in a fully dynamic graph with worst-case update times $${\hat{O}}(\sqrt{n})$$ O ^ ( n ) and $${\tilde{O}}(1)$$ O ~ ( 1 ) respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio $$(2 - \delta )$$ ( 2 - δ ) (for any $$\delta > 0$$ δ > 0 ) and $$(2 + \epsilon )$$ ( 2 + ϵ ) were both shown by Roghani et al. (Beating the folklore algorithm for dynamic matching, 2021) with update times $$O(n^{3/4})$$ O ( n 3 / 4 ) and $$O_\epsilon (\sqrt{n})$$ 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 $$O_\epsilon (\sqrt{n})$$ O ϵ ( n ) and $${\tilde{O}}(1)$$ 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 $$(3/2 + \epsilon )$$ ( 3 / 2 + ϵ ) 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 $$(\alpha , \delta )$$ ( α , δ ) -approximate matching sparsifier if at all times H satisfies that $$\mu (H) \cdot \alpha + \delta \cdot n \ge \mu (G)$$ μ ( H ) · α + δ · n ≥ μ ( G ) (define $$(\alpha , \delta )$$ ( α , δ ) -approximation similarly for matchings). We show how to maintain a locally damaged version of the EDCS which is a $$(3/2 + \epsilon , \delta )$$ ( 3 / 2 + ϵ , δ ) -approximate matching sparsifier. We further show how to reduce the maintenance of an $$\alpha $$ α -approximate maximum matching to the maintenance of an $$(\alpha , \delta )$$ ( α , δ ) -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 $${\hat{O}}(1)$$ O ^ ( 1 ) or $${\tilde{O}}(1)$$ O ~ ( 1 ) and is deterministic or randomized against an adaptive adversary respectively. To achieve $$(2 + \epsilon )$$ ( 2 + ϵ ) -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.