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

TL;DR

This paper introduces a randomized reduction that transforms fractional matching algorithms into integral ones, leading to the first efficient worst-case update time algorithms for approximate maximum weight matchings.

Contribution

It presents a novel reduction from integral to fractional algorithms, enabling the first constant-approximate worst-case polylogarithmic update time maximum weight matching algorithm.

Findings

Achieved a randomized $(2+)$-approximate integral matching algorithm with polylog worst-case update time.

First to break polynomial worst-case update time barrier for approximate matchings.

Established a new baseline for dynamic maximum weight matching algorithms.

Abstract

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 \emph{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.