Dynamic Matching: Reducing Integral Algorithms to Approximately-Maximal Fractional Algorithms

We present a simple randomized reduction from fully-dynamic integral matching algorithms to fully-dynamic “approximately-maximal” fractional matching algorithms. Applying this reduction to the recent fractional matching algorithm of Bhattacharya, Henzinger, and Nanongkai (SODA 2017), we obtain a novel result for the integral problem. Specifically, our main result is a randomized fully-dynamic (2+ ǫ)-approximate integral matching algorithm with small polylog worst-case update time. For the (2 + ǫ)-approximation regime only a fractional fully-dynamic (2 + ǫ)-matching algorithm with worst-case polylog update time was previously known, due to Bhattacharya et al. (SODA 2017). Our algorithm is the first algorithm that maintains approximate matchings with worst-case update time better than polynomial, for any constant approximation ratio. ∗This work was done in part while these authors were visiting the Simons Institute for the Theory of Computing. Cliff Stein’s work is partly supported by NSF Grant CCF-1421161.