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 [...]

翻译：动态集合覆盖问题近年来受到越来越多的研究关注。在该问题中，我们接收一个至多包含 $n$ 个元素的动态全集和一个包含 $m$ 个集合的固定族作为输入，其中每个元素至多出现在 $f$ 个集合中，每个集合的代价在 $[1/C, 1]$ 区间内，目标是在元素动态更新的情况下高效地维护一个近似最小集合覆盖。目前已知两种将经典贪心算法动态化的算法，分别以 $O(f \log n)$ 和 $O(\frac{f \log n}{\epsilon^5})$ 的摊销更新时间提供了 $O(\log n)$ 和 $((1+\epsilon)\ln n)$ 的近似比 [GKKP (STOC'17); SU (STOC'23)]。能否以较低的最坏情况更新时间获得 $O(\log n)$（甚至更差）近似比的问题一直悬而未决——即使对于无权实例，目前也只知朴素的 $O(f \cdot n)$ 时间界。在本工作中，我们设计了第一个适用于高效去摊销化的摊销贪心算法，并开发了一种适用于集合覆盖问题的无损去摊销化方法，二者结合产生了一个具有 $O(\frac{f\log n}{\epsilon^2})$ 最坏情况更新时间的 $((1+\epsilon)\ln n)$ 近似算法。我们的最坏情况时间界——首个突破朴素 $O(f \cdot n)$ 界的成果——匹配了先前最佳的摊销界，并且实际上改进了其对 $\epsilon$ 的依赖关系。此外，为了证明我们提出的去摊销化方法的适用性，我们将其与 [BHN (FOCS'19)] 的原对偶摊销算法结合使用，获得了一个具有 $O(\frac{f\log n}{\epsilon^2})$ 最坏情况更新时间的 $((1+\epsilon)f)$ 近似算法，改进了先前的最佳界 $O(\frac{f \cdot \log^2(Cn)}{\epsilon^3})$ [BHNW (SODA'21)]。最后，作为我们的集合覆盖结果直接带来的影响，我们 [...]