Maintaining an EDCS in General Graphs: Simpler, Density-Sensitive and with Worst-Case Time Bounds
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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 O_ϵ(m^1/4); we use the O_ϵ 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 O_ϵ(m^1/4), 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 O_ϵ(m^1/4), 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 approximation slightly below 2 [Behnezhad, Lacki and Mirrokni, SODA'20]. In this work we[quasi nanos, gigantium humeris insidentes] 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_ϵ(m^1/4) 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 O_ϵ(√(α)).
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