  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 amortised update time of $O(1)$ with
high probability. This algorithm requires the assumption of an {\em 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 {\em deterministic} fully dynamic algorithm that maintains a
$O(1)$-approximate minimum vertex cover and maximum fractional matching, with
an amortised 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 amortised
update time of $O(\log n/\epsilon^2)$. Our results also extend to a fully
dynamic $O(f^3)$-approximate algorithm with $O(f^2)$ amortized update time for
the hypergraph vertex cover and fractional hypergraph matching problems, where
every hyperedge has at most $f$ vertices.
