The minimum set cover (MSC) problem admits two classic algorithms: a greedy $\ln n$-approximation and a primal-dual $f$-approximation, where $n$ is the universe size and $f$ is the maximum frequency of an element. Both algorithms are simple and efficient, and remarkably -- one cannot improve these approximations under hardness results by more than a factor of $(1+\epsilon)$, for any constant $\epsilon > 0$. In their pioneering work, Gupta et al. [STOC'17] showed that the greedy algorithm can be dynamized to achieve $O(\log n)$-approximation with update time $O(f \log n)$. Building on this result, Hjuler et al. [STACS'18] dynamized the greedy minimum dominating set (MDS) algorithm, achieving a similar approximation with update time $O(\Delta \log n)$ (the analog of $O(f \log n)$), albeit for unweighted instances. The approximations of both algorithms, which are the state-of-the-art, exceed the static $\ln n$-approximation by a rather large constant factor. In sharp contrast, the current best dynamic primal-dual MSC algorithms achieve fast update times together with an approximation that exceeds the static $f$-approximation by a factor of (at most) $1+\epsilon$, for any $\epsilon > 0$. This paper aims to bridge the gap between the best approximation factor of the dynamic greedy MSC and MDS algorithms and the static $\ln n$ bound. We present dynamic algorithms for weighted greedy MSC and MDS with approximation $(1+\epsilon)\ln n$ for any $\epsilon > 0$, while achieving the same update time (ignoring dependencies on $\epsilon$) of the best previous algorithms (with approximation significantly larger than $\ln n$). Moreover, [...]

翻译：最小集合覆盖（MSC）问题存在两种经典算法：贪心$\ln n$近似算法和原始-对偶$f$近似算法，其中$n$为全域规模，$f$为元素的最大出现频率。这两种算法简洁高效，且值得注意的是——在硬度结论下，对于任意常数$\epsilon > 0$，这些近似比无法改进超过$(1+\epsilon)$倍。Gupta等人[STOC'17]的开创性工作表明，贪心算法可被动态化实现$O(\log n)$近似比，更新时间为$O(f \log n)$。基于此结果，Hjuler等人[STACS'18]将贪心最小支配集（MDS）算法动态化，针对非加权实例实现了类似近似比（对应$O(f \log n)$），更新时间为$O(\Delta \log n)$。这两种目前最优算法的近似比均显著高于静态$\ln n$近似常系数倍。与此形成鲜明对比的是，当前最优动态原始-对偶MSC算法在实现快速更新时间的同时，其近似比比静态$f$近似比最多超出$(1+\epsilon)$倍（对任意$\epsilon > 0$）。本文旨在弥合动态贪心MSC与MDS算法的最优近似比与静态$\ln n$界限之间的差距。我们针对加权贪心MSC和MDS问题提出了动态算法，对于任意$\epsilon > 0$可实现$(1+\epsilon)\ln n$近似比，同时达到与先前最优算法相同的更新时间复杂度（忽略对$\epsilon$的依赖），而先前算法的近似比远大于$\ln n$。此外，[...]