  The dynamic set cover problem has been subject to growing research attention
in recent years. In this problem, we are given as input a dynamic universe of
at most $n$ elements and a fixed collection of $m$ sets, where each element
appears in a most $f$ sets and the cost of each set is in $[1/C, 1]$, and the
goal is to efficiently maintain an approximate minimum set cover under element
updates.
  Two algorithms that dynamize the classic greedy algorithm are known,
providing $O(\log n)$ and $((1+\epsilon)\ln n)$-approximation with amortized
update times $O(f \log n)$ and $O(\frac{f \log n}{\epsilon^5})$, respectively
[GKKP (STOC'17); SU (STOC'23)]. The question of whether one can get
approximation $O(\log n)$ (or even worse) with low worst-case update time has
remained open -- only the naive $O(f \cdot n)$ time bound is known, even for
unweighted instances.
  In this work we devise the first amortized greedy algorithm that is amenable
to an efficient deamortization, and also develop a lossless deamortization
approach suitable for the set cover problem, the combination of which yields a
$((1+\epsilon)\ln n)$-approximation algorithm with a worst-case update time of
$O(\frac{f\log n}{\epsilon^2})$. Our worst-case time bound -- the first to
break the naive $O(f \cdot n)$ bound -- matches the previous best amortized
bound, and actually improves its $\epsilon$-dependence.
  Further, to demonstrate the applicability of our deamortization approach, we
employ it, in conjunction with the primal-dual amortized algorithm of [BHN
(FOCS'19)], to obtain a $((1+\epsilon)f)$-approximation algorithm with a
worst-case update time of $O(\frac{f\log n}{\epsilon^2})$, improving over the
previous best bound of $O(\frac{f \cdot \log^2(Cn)}{\epsilon^3})$ [BHNW
(SODA'21)].
  Finally, as direct implications of our results for set cover, we [...]
