We introduce the problem of finding a set $B$ of $k$ points in $[0,1]^n$ such that the expected cost of the cheapest point in $B$ that dominates a random point from $[0,1]^n$ is minimized. We study the case where the coordinates of the random points are independently distributed and the cost function is linear. This problem arises naturally in various application areas where customers' requests are satisfied based on predefined products, each corresponding to a subset of features. We show that the problem is NP-hard already for $k=2$ when each coordinate is drawn from $\{0,1\}$, and obtain an FPTAS for general fixed $k$ under mild assumptions on the distributions.

翻译：我们提出了在$[0,1,1]美元中找到一套以美元计点的固定汇率(单位:0,1美元)的问题,以便尽可能降低以美元计点的最廉价点(单位:$[0,1,1美元])的预期成本,我们研究随机点的坐标独立分布和成本函数线性的情况,这个问题自然出现在各种应用领域,在这些应用领域,客户的要求根据预先确定的产品得到满足,每个产品都与一系列特征相对应。我们发现,当每个坐标从$0,1美元中抽取时,问题已经以美元=2美元的形式出现,并且根据对分配的轻度假设,获得一般固定美元的FPTAS。