Fully Dynamic $(1+\epsilon)$-Approximate Matchings

TL;DR

This paper introduces a novel data structure that maintains near-optimal maximum matchings in sparse graphs with sublinear update time, improving approximation ratios and extending to weighted matchings.

Contribution

The paper presents the first dynamic data structures achieving near-optimal matchings with sublinear update time, including extensions to weighted graphs with better approximation guarantees.

Findings

Maintains a (1+ε)-approximate maximum matching in O(√m ε^{-2}) time per update.

Extends approach to weighted matchings with O(√m log N) time per update.

Improves previous approximation ratios for dynamic weighted matchings.

Abstract

We present the first data structures that maintain near optimal maximum cardinality and maximum weighted matchings on sparse graphs in sublinear time per update. Our main result is a data structure that maintains a $(1+\epsilon)$ approximation of maximum matching under edge insertions/deletions in worst case $O(\sqrt{m}\epsilon^{-2})$ time per update. This improves the 3/2 approximation given in [Neiman,Solomon,STOC 2013] which runs in similar time. The result is based on two ideas. The first is to re-run a static algorithm after a chosen number of updates to ensure approximation guarantees. The second is to judiciously trim the graph to a smaller equivalent one whenever possible. We also study extensions of our approach to the weighted setting, and combine it with known frameworks to obtain arbitrary approximation ratios. For a constant $\epsilon$ and for graphs with edge weights between 1 and N, we design an algorithm that maintains an $(1+\epsilon)$-approximate maximum weighted matching in $O(\sqrt{m} \log N)$ time per update. The only previous result for maintaining weighted matchings on dynamic graphs has an approximation ratio of 4.9108, and was shown in [Anand,Baswana,Gupta,Sen, FSTTCS 2012, arXiv 2012].