  In the dynamic minimum set cover problem, a challenge is to minimize the
update time while guaranteeing close to the optimal $\min(O(\log n), f)$
approximation factor. (Throughout, $m$, $n$, $f$, and $C$ are parameters
denoting the maximum number of sets, number of elements, frequency, and the
cost range.) In the high-frequency range, when $f=\Omega(\log n)$, this was
achieved by a deterministic $O(\log n)$-approximation algorithm with $O(f \log
n)$ amortized update time [Gupta et al. STOC'17]. In the low-frequency range,
the line of work by Gupta et al. [STOC'17], Abboud et al. [STOC'19], and
Bhattacharya et al. [ICALP'15, IPCO'17, FOCS'19] led to a deterministic
$(1+\epsilon)f$-approximation algorithm with $O(f \log (Cn)/\epsilon^2)$
amortized update time. In this paper we improve the latter update time and
provide the first bounds that subsume (and sometimes improve) the
state-of-the-art dynamic vertex cover algorithms. We obtain:
  1. $(1+\epsilon)f$-approximation ratio in $O(f\log^2 (Cn)/\epsilon^3)$
worst-case update time: No non-trivial worst-case update time was previously
known for dynamic set cover. Our bound subsumes and improves by a logarithmic
factor the $O(\log^3 n/\text{poly}(\epsilon))$ worst-case update time for
unweighted dynamic vertex cover (i.e., when $f=2$ and $C=1$) by Bhattacharya et
al. [SODA'17].
  2. $(1+\epsilon)f$-approximation ratio in
$O\left((f^2/\epsilon^3)+(f/\epsilon^2) \log C\right)$ amortized update time:
This result improves the previous $O(f \log (Cn)/\epsilon^2)$ update time bound
for most values of $f$ in the low-frequency range, i.e. whenever $f=o(\log n)$.
It is the first that is independent of $m$ and $n$. It subsumes the constant
amortized update time of Bhattacharya and Kulkarni [SODA'19] for unweighted
dynamic vertex cover (i.e., when $f = 2$ and $C = 1$).
