Improved Algorithm for Dynamic b-Matching.

Recently there has been extensive work on maintaining (approximate) maximum matchings in dynamic graphs. We consider a generalisation of this problem known as the maximum b-matching: Every node v has a positive integral capacity b_v, and the goal is to maintain an (approximate) maximum-cardinality subset of edges that contains at most b_v edges incident on every node v. The maximum matching problem is a special case of this problem where b_v = 1 for every node v.Bhattacharya, Henzinger and Italiano [ICALP 2015] showed how to maintain a O(1) approximate maximum b-matching in a graph in O(log^3 n) amortised update time. Their approximation ratio was a large (double digit) constant. We significantly improve their result both in terms of approximation ratio as well as update time. Specifically, we design a randomised dynamic algorithm that maintains a (2+epsilon)-approximate maximum $b$-matching in expected amortised O(1/epsilon^4) update time. Thus, for every constant epsilon in (0, 1), we get expected amortised O(1) update time. Our algorithm generalises the framework of Baswana, Gupta, Sen [FOCS 2011] and Solomon [FOCS 2016] for maintaining a maximal matching in a dynamic graph.