Deterministic Dynamic Matching in O(1) Update Time

Sayan Bhattacharya, Deeparnab Chakrabarty, Monika Henzinger
2019 Algorithmica  
We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld (in: Proceedings of the ACM symposium on theory of computing (STOC), 2010), this problem has received significant attention in recent years. Very recently, extending the framework of Baswana et al. (in: gave a randomized dynamic algorithm for this problem that
more » ... an approximation ratio of 2 and an amortized update time of O(1) with high probability. This algorithm requires the assumption of an oblivious adversary, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm's past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in O(1) update time. In this paper, we resolve this question. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya et al. (in: Proceedings of the ACM-SIAM symposium on discrete algorithms (SODA), 2015); it had an approximation ratio of (2 + ε) and an amortized update time of O(log n/ε 2 ). Our result can be generalized to give a fully dynamic O( f 3 )-approximate algorithm with O( f 2 ) amortized update time for the hypergraph vertex cover and fractional hypergraph matching problem, where every hyperedge has at most f vertices.
doi:10.1007/s00453-019-00630-4 fatcat:vqgsq4gghjablm43jzq75nlho4