Maintaining an EDCS in General Graphs: Simpler, Density-Sensitive and with Worst-Case Time Bounds

In their breakthrough ICALP’15 paper, Bernstein and Stein presented an algorithm for maintaining a (3/2 + ε)-approximate maximum matching in fully dynamic bipartite graphs with a worst-case update time of Oε(m¹/⁴); we use the Oε notation to suppress the ε-dependence. Their main technical contribution was in presenting a new type of bounded-degree subgraph, which they named an edge degree constrained subgraph (EDCS), which contains a large matching — of size that is smaller than the maximum matching size of the entire graph by at most a factor of 3/2+ε. They demonstrate that the EDCS can be maintained with a worst-case update time of Oε(m¹/⁴), and their main result follows as a direct corollary. In their followup SODA’16 paper, Bernstein and Stein generalized their result for general graphs, achieving the same update time of Oε(m¹/⁴), albeit with an amortized rather than worst-case bound. To date, the best deterministic worst-case update time bound for any better-than-2 approximate matching is O(√m) [Neiman and Solomon, STOC’13], [Gupta and Peng, FOCS’13]; allowing randomization (against an oblivious adversary) one can achieve a much better (still polynomial) update time for an approximation slightly below 2 [Behnezhad, Lacki and Mirrokni, SODA’20]. In this work we¹ simplify the approach of Bernstein and Stein for bipartite graphs, which allows us to generalize it for general graphs while maintaining the same bound of Oε(m¹/⁴) on the worst-case update time. Moreover, our approach is density-sensitive: If the arboricity of _the dynamic graph is bounded by α at all times, then the worst-case update time of the algorithm is Oε(√α). Recent related work: Independently and concurrently to our work, Roghani, Saberi and Wajc [arXiv’21] obtained two dynamic algorithms for approximate maximum matching with worst-case update time bounds. Their first algorithm achieves approximation factor slightly better than 2 within O(√n · m¹/⁸) update time, and their second algorithm achieves approximation factor (2 + ε) for any ε > 0 within Oε(√n) update time. In terms of techniques, the two works are entirely disjoint.