New Trade-Offs for Fully Dynamic Matching via Hierarchical EDCS

We study the maximum matching problem in fully dynamic graphs: a graph is undergoing both edge insertions and deletions, and the goal is to efficiently maintain a large matching after each edge update. This problem has received considerable attention in recent years. The known algorithms naturally exhibit a trade-off between the quality of the matching maintained (i.e., the approximation ratio) and the time needed per update. While several interesting results have been obtained, the optimal behavior of this trade-off remains largely unclear. Our main contribution is a new approach to designing fully dynamic approximate matching algorithms that in a unified manner not only (essentially) recovers all previously known trade-offs that were achieved via very different techniques, but reveals some new ones as well.

Specifically, we introduce a generalization of the edge-degree constrained subgraph (EDCS) of Bernstein and Stein (2015) that we call the hierarchical EDCS (HEDCS). We also present a randomized algorithm for efficiently maintaining an HEDCS. In an m-edge graph with maximum degree ∆, for any integer k ≥ 0 that is essentially the number of levels of the hierarchy in HEDCS, our algorithm takes O(min∆ 1/(k+1) , m 1/(2k+2) ) worst-case update-time and maintains an (almost) α(k)-approximate matching where we show:

)) for any δ > 0, and α(log ∆) ≥ 1 2 . These bounds recover all previous trade-offs known for dynamic matching in the literature up to logarithmic factors in the update-time.

• α(2) > .612 for bipartite graphs, and α(2) > .609 for general graphs.

Note that these approximations are obtained in O(min∆ 1/3 , m 1/6 ) update-time.

• α(3) > .563 for bipartite graphs, and α(3) > .532 for general graphs. Note that these approximations are obtained in O(min∆ 1/4 , m 1/8 ) update-time.