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 + )-approximate maximum matching in fully dynamic bipartite graphs with a worst-case update time of O (m ); we use the O notation to suppress the -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 + . They demonstrate that the EDCS can be maintained with a worst-case update time of O (m ), 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 O (m ), 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 approximation slightly below 2 [Behnezhad, Lacki and Mirrokni, SODA’20]. In this work we 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 O (m ) 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 O ( √ 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 ·m) update time, and their second algorithm achieves approximation factor (2 + ) for any > 0 within O ( √ n) update time. In terms of techniques, the two works are entirely disjoint. Research was partially supported by Israel Science Foundation (ISF) grant 1991/19, and by a grant from the United States-Israel Binational Science Foundation (BSF) and the United States National Science Foundation (NSF). quasi nanos, gigantium humeris insidentes ar X iv :2 10 8. 08 82 5v 1 [ cs .D S] 1 9 A ug 2 02 1