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+epsilon)-approximate integral matching algorithm with small polylog worst-case update time. For the (2+epsilon)-approximation regime only a fractional fully-dynamic (2+epsilon)-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. As a consequence, we also obtain the first constant-approximate worst-case polylogarithmic update time maximum weight matching algorithm.