The dynamic set cover problem has been subject to extensive research since the pioneering works of [Bhattacharya et al, 2015] and [Gupta et al, 2017]. The input is a set system $(U, S)$ on a fixed collection $S$ of sets and a dynamic universe of elements, where each element appears in a most $f$ sets and the cost of each set lies in the range $[1/C, 1]$, and the goal is to efficiently maintain an approximately-minimum set cover under insertions and deletions of elements. Most previous work considers the low-frequency regime, namely $f = O(\log n)$, and this line of work has culminated with a deterministic $(1+\epsilon)f$-approximation algorithm with amortized update time $O(\frac{f^2}{\epsilon^3} + \frac{f}{\epsilon^2}\log C)$ [Bhattacharya et al, 2021]. In the high-frequency regime of $f = \Omega(\log n)$, an $O(\log n)$-approximation algorithm with amortized update time $O(f\log n)$ was given by [Gupta et al, 2017]. Interestingly, at the intersection of the two regimes, i.e., $f = \Theta(\log n)$, the state-of-the-art results coincide: approximation $\Theta(f) = \Theta(\log n)$ with amortized update time $O(f^2) = O(f \log n) = O(\log^2 n)$. Up to this date, no previous work achieved update time of $o(f^2)$. In this paper we break the $\Omega(f^2)$ update time barrier via the following results: (1) $(1+\epsilon)f$-approximation can be maintained in $O\left(\frac{f}{\epsilon^3}\log^*f + \frac{f}{\epsilon^3}\log C\right) = O_{\epsilon,C}(f \log^* f)$ expected amortized update time; our algorithm works against an adaptive adversary. (2) $(1+\epsilon)f$-approximation can be maintained deterministically in $O\left(\frac{1}{\epsilon}f\log f + \frac{f}{\epsilon^3} + \frac{f\log C}{\epsilon^2}\right) = O_{\epsilon,C}(f \log f)$ amortized update time.

翻译：动态集合覆盖问题自[Bhattacharya等人，2015]和[Gupta等人，2017]的开创性工作以来一直是广泛研究的对象。输入是一个集合系统$(U, S)$，其中$S$是固定的集合集合，元素构成动态全集，每个元素最多出现在$f$个集合中，且每个集合的成本在$[1/C, 1]$范围内，目标是在元素插入和删除操作下高效维护近似最小的集合覆盖。以往的大多数研究考虑低频率情形，即$f = O(\log n)$，这一研究方向已取得最终成果：一个确定性$(1+\epsilon)f$近似算法，其均摊更新时间为$O(\frac{f^2}{\epsilon^3} + \frac{f}{\epsilon^2}\log C)$ [Bhattacharya等人，2021]。在$f = \Omega(\log n)$的高频率情形中，[Gupta等人，2017]提出了一个$O(\log n)$近似算法，其均摊更新时间为$O(f\log n)$。有趣的是，在两个情形的交汇点，即$f = \Theta(\log n)$时，现有最优结果一致：近似比为$\Theta(f) = \Theta(\log n)$，均摊更新时间为$O(f^2) = O(f \log n) = O(\log^2 n)$。迄今为止，尚无先前工作实现$o(f^2)$的更新时间。在本文中，我们通过以下结果突破了$\Omega(f^2)$的更新时间障碍：(1) $(1+\epsilon)f$近似可以在期望均摊更新时间$O\left(\frac{f}{\epsilon^3}\log^*f + \frac{f}{\epsilon^3}\log C\right) = O_{\epsilon,C}(f \log^* f)$内维护；我们的算法能对抗自适应敌手。(2) $(1+\epsilon)f$近似可以在确定性均摊更新时间$O\left(\frac{1}{\epsilon}f\log f + \frac{f}{\epsilon^3} + \frac{f\log C}{\epsilon^2}\right) = O_{\epsilon,C}(f \log f)$内维护。