  We present a deterministic dynamic algorithm for maintaining a
$(1+\epsilon)f$-approximate minimum cost set cover with
$O(f\log(Cn)/\epsilon^2)$ amortized update time, when the input set system is
undergoing element insertions and deletions. Here, $n$ denotes the number of
elements, each element appears in at most $f$ sets, and the cost of each set
lies in the range $[1/C, 1]$. Our result, together with that of Gupta et al.
[STOC`17], implies that there is a deterministic algorithm for this problem
with $O(f\log(Cn))$ amortized update time and $O(\min(\log n,
f))$-approximation ratio, which nearly matches the polynomial-time hardness of
approximation for minimum set cover in the static setting. Our update time is
only $O(\log (Cn))$ away from a trivial lower bound.
  Prior to our work, the previous best approximation ratio guaranteed by
deterministic algorithms was $O(f^2)$, which was due to Bhattacharya et al.
[ICALP`15]. In contrast, the only result that guaranteed $O(f)$-approximation
was obtained very recently by Abboud et al. [STOC`19], who designed a dynamic
algorithm with $(1+\epsilon)f$-approximation ratio and $O(f^2 \log n/\epsilon)$
amortized update time. Besides the extra $O(f)$ factor in the update time
compared to our and Gupta et al.'s results, the Abboud et al. algorithm is
randomized, and works only when the adversary is oblivious and the sets are
unweighted (each set has the same cost).
  We achieve our result via the primal-dual approach, by maintaining a
fractional packing solution as a dual certificate. Unlike previous primal-dual
algorithms that try to satisfy some local constraints for individual sets at
all time, our algorithm basically waits until the dual solution changes
significantly globally, and fixes the solution only where the fix is needed.
