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

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 + ǫ) approximation of maximum matching under edge insertions/deletions in worst case O( √ mǫ) 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 ǫ and for graphs with edge weights between 1 and N , we design an algorithm that maintains an (1 + ǫ)-approximate maximum weighted matching in O( √ m logN) 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].