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

Abstract

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 [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 .