We consider a large family of problems in which an ordering (or, more precisely, a chain of subsets) of a finite set must be chosen to minimize some weighted sum of costs. This family includes variations of Min Sum Set Cover (MSSC), several scheduling and search problems, and problems in Boolean function evaluation. We define a new problem, called the Min Sum Ordering Problem (MSOP) which generalizes all these problems using a cost and a weight function defined on subsets of a finite set. Assuming a polynomial time $\alpha$-approximation algorithm for the problem of finding a subset whose ratio of weight to cost is maximal, we show that under very minimal assumptions, there is a polynomial time $4 \alpha$-approximation algorithm for MSOP. This approximation result generalizes a proof technique used for several distinct problems in the literature. We apply this to obtain a number of new approximation results.

翻译：我们考虑的是一个大系列问题,即必须选择一个定序(或更确切地说,一个子集链)来尽量减少某些加权成本总和。这个系列包括 Min Sum Set Cover (MSS) 的变异、 几个排期和搜索问题, 以及布林函数评估中的问题。 我们定义了一个新问题, 叫做 Min Sum 命令问题 (MSOP ), 使用一个成本和一定数组子集界定的权重函数来概括所有这些问题。 假设要找到一个其重量与成本之比最高的一个子, 我们用这个算法来获取一些新的近似结果。