  In the (fully) dynamic set cover problem, we have a collection of $m$ sets
from a universe of size $n$ that undergo element insertions and deletions; the
goal is to maintain an approximate set cover of the universe after each update.
We give an $O(f^2)$ update time algorithm for this problem that achieves an
$f$-approximation, where $f$ is the maximum number of sets that an element
belongs to; under the unique games conjecture, this approximation is best
possible for any fixed $f$. This is the first algorithm for dynamic set cover
with approximation ratio that {exactly} matches $f$ (as opposed to {almost} $f$
in prior work), as well as the first one with runtime \emph{independent of
$n,m$} (for any approximation factor of $o(f^3)$).
  Prior to our work, the state-of-the-art algorithms for this problem were
$O(f^2)$ update time algorithms of Gupta et al. [STOC'17] and Bhattacharya et
al. [IPCO'17] with $O(f^3)$ approximation, and the recent algorithm of
Bhattacharya et al. [FOCS'19] with $O(f \cdot \log{n}/\epsilon^2)$ update time
and $(1+\epsilon) \cdot f$ approximation, improving the $O(f^2 \cdot
\log{n}/\epsilon^5)$ bound of Abboud et al. [STOC'19].
  The key technical ingredient of our work is an algorithm for maintaining a
{maximal} matching in a dynamic hypergraph of rank $r$, where each hyperedge
has at most $r$ vertices, which undergoes hyperedge insertions and deletions in
$O(r^2)$ amortized update time; our algorithm is randomized, and the bound on
the update time holds in expectation and with high probability. This result
generalizes the maximal matching algorithm of Solomon [FOCS'16] with constant
update time in ordinary graphs to hypergraphs, and is of independent merit; the
previous state-of-the-art algorithms for set cover do not translate to
(integral) matchings for hypergraphs, let alone a maximal one. Our quantitative
result for the set cover problem is [...]
