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

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 [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 dynamic algorithm for this problem that has 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, Henzinger and Italiano [SODA 2015]; it had an approximation ratio of (2+ ε) and an amortized update time of O(logn/ε2). 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. ∗University of Warwick, UK. Email: s.bhattacharya@warwick.ac.uk †Department of Computer Science, Dartmouth College, 6211 Sudikoff Lab, Hanover, NH 03755, USA. Email: deeparnab.chakrabarty@dartmouth.edu. Work done while the author was at Microsoft Research, India. ‡University of Vienna, Austria. Email: monika.henzinger@univie.ac.at.The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/20072013) / ERC Grant Agreement no. 340506.