A set cover of a hypergraph $H$ is a set of vertices intersecting every hyperedge. In the minimum sum set cover problem, vertices are selected one by one; each edge pays the position of the first vertex that hits it, and the objective is to minimize the total cost. When $H$ is a graph, this is the minimum sum vertex cover problem. A solution is specified by a set cover $S$ together with an ordering of its vertices. While the classical set cover problem seeks to minimize $|S|$, the minimum sum variant favors covering many edges early and may prefer larger covers. This motivates a natural question: how large can the gap between~$\overrightarrowτ$ and $τ$ be? We prove an upper bound $\overrightarrowτ \le τ\log_{2} \lvert E(H)\rvert$, and show that for any positive~$n$, there exists a hypergraph $H$ on $n + 3$ vertices with $τ=3$ and $\overrightarrowτ=n$. For graphs, we obtain stronger bounds: we prove~$\overrightarrowτ \le 2τ\log_{2} τ$, improving the bound of Liu et al.\ [Theor. Comput. Sci., 2025], and we construct graphs with~$\overrightarrowτ = Ω\left( \frac{τ\log τ}{\log\log τ}\right)$, nearly matching this upper bound. On the algorithmic side, we show that minimum sum set cover is fixed-parameter tractable on bounded-rank hypergraphs, parameterized by~$\overrightarrowτ$, extending the algorithm of Liu et al.\ for graphs (i.e., rank-two hypergraphs).

翻译：超图$H$的集覆盖是指与每条超边相交的顶点集。在最小和集覆盖问题中，顶点被逐一选取；每条超边需支付覆盖它的首个顶点的位置，目标是最小化总成本。当$H$为图时，该问题即最小和顶点覆盖问题。一个解由集覆盖$S$及其顶点的排序共同定义。经典集覆盖问题旨在最小化$|S|$，而最小和变体偏好尽早覆盖更多超边，可能选择更大的覆盖。这引出一个自然问题：$\overrightarrowτ$与$τ$之间的差距可能有多大？我们证明上界$\overrightarrowτ \le τ\log_{2} \lvert E(H)\rvert$，并说明对任意正整数$n$，存在顶点数为$n+3$且满足$τ=3$、$\overrightarrowτ=n$的超图$H$。对于图，我们得到更强界：证明$\overrightarrowτ \le 2τ\log_{2} τ$，改进了Liu等人[Theor. Comput. Sci., 2025]的界，并构造满足$\overrightarrowτ = Ω\left( \frac{τ\log τ}{\log\log τ}\right)$的图，几乎匹配该上界。在算法方面，我们证明最小和集覆盖在有界秩超图上关于参数$\overrightarrowτ$是固定参数可解的，将Liu等人针对图（即秩为二的超图）的算法进行了推广。