Fully Dynamic (1+ e)-Approximate 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 + ǫ) approximation of maximum matching under edge insertions/deletions in worst case O( √ mǫ −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 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 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].

New deterministic approximation algorithms for fully dynamic matching

“…In fact, our approximation guarantee almost matches the best (2-approximation) one provided by a randomized algorithm [2]. 2 Our algorithm for Theorem 1.1 is obtained by combining previous techniques [5,7,15] with two new ideas that concern fractional matchings. First, we dynamize the degree splitting process previously used in the parallel and distributed algorithms literature [9] and use it to reduce the size of the support of the fractional matching maintained by the algorithm of [5].…”

“…First, we dynamize the degree splitting process previously used in the parallel and distributed algorithms literature [9] and use it to reduce the size of the support of the fractional matching maintained by the algorithm of [5]. This helps us maintain an approximate integral matching cheaply using the result in [7]. This idea alone already leads to a (3 + ε)-approximation deterministic algorithm.…”

“…Gupta and Peng [7] gave a dynamic algorithm that maintains a (1 + ε)-approximate maximum matching in O( √ m/ε 2 ) update time. A simple modification of their algorithm gives the following result.…”

Big data algorithms beyond machine learning

“…One, therefore, asks which static problems solvable in time f(m) can be fully "dynamized", in the sense of having dynamic algorithms that support updates in O(f(m) / m) time. This question has been answered affirmatively for many fundamental graph problems including connectivity (e.g., [30,33,34,52]), reachability [32], shortest paths (e.g., [8,18,31]), and maximum matching [9,27,49].…”

Maintaining Near-Popular Matchings

“…Baswana et al [7] provide a randomized 2-approximation in O(log n) amortized time. For deterministic algorithms, Gupta and Peng [17] gave a (1 + ε)-approximation in O( √ m/ε 2 ) worst-case update time. Bhattacharya et al [9] showed a deterministic (4 + ε)-approximation in O(m 1/3 /ε 2 ) worst-case update time.…”