We study the \emph{generalized min-sum set cover} (GMSSC) problem, where given a collection of hyperedges $E$ with arbitrary covering requirements $\{k_e \in \mathbb{Z}^+ : e \in E\}$, the objective is to find an ordering of the vertices that minimizes the total cover time of the hyperedges. A hyperedge $e$ is considered covered at the first time when $k_e$ of its vertices appear in the ordering. We present a $4.509$-approximation algorithm for GMSSC, improving upon the previous best-known guarantee of $4.642$~\cite[SODA'21]{BansalBFT21}. Our approach retains the general LP-based framework of Bansal, Batra, Farhadi, and Tetali~\cite{BansalBFT21} but provides an improved analysis that narrows the gap toward the lower bound of $4$-approximation assuming P$\neq$NP. Our analysis takes advantage of the constraints of the linear program in a nontrivial way, along with new lower-tail bounds for the sums of independent Bernoulli random variables, which could be of independent interest.

翻译：我们研究广义最小和集合覆盖（GMSSC）问题，其中给定一组具有任意覆盖需求$\{k_e \in \mathbb{Z}^+ : e \in E\}$的超边集合$E$，目标是找到一个顶点排序，使得所有超边的总覆盖时间最小化。当超边$e$在排序中出现$k_e$个顶点时，该超边被视为首次被覆盖。我们提出了GMSSC的4.509近似算法，改进了此前最优的4.642保证~\cite[SODA'21]{BansalBFT21}。我们的方法沿用了Bansal、Batra、Farhadi和Tetali提出的基于线性规划的基本框架~\cite{BansalBFT21}，但通过改进分析缩小了与假设P≠NP下4近似下界之间的差距。我们的分析非平凡地利用了线性规划约束，并结合了独立伯努利随机变量和的新的下尾界——这一结果可能具有独立的研究价值。