Deterministic Fully Dynamic Approximate Vertex Cover and Fractional Matching in O(1) Amortized Update Time

Abstract

We consider the problems of maintaining approximate maximum matching and minimum vertex cover in a dynamic graph. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized 2-approximation dynamic algorithm for this problem that has amortized update time of O(1) with high probability. We consider the natural open question of derandomizing this result. 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, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of $$(2+\epsilon )$$ and an amortized update time of $$O(\log n/\epsilon ^2)$$ . Our result can be generalized to give a fully dynamic $$O(f^3)$$ -approximation algorithm with $$O(f^2)$$ amortized update time for the hypergraph vertex cover and fractional matching problems, where every hyperedge has at most f vertices.