We consider the problem of sketching a set valuation function, which is defined as the expectation of a valuation function of independent random item values. We show that for monotone subadditive or submodular valuation functions satisfying a weak homogeneity condition, or certain other conditions, there exist discretized distributions of item values with $O(k\log(k))$ support sizes that yield a sketch valuation function which is a constant-factor approximation, for any value query for a set of items of cardinality less than or equal to $k$. The discretized distributions can be efficiently computed by an algorithm for each item's value distribution separately. Our results hold under conditions that accommodate a wide range of valuation functions arising in applications, such as the value of a team corresponding to the best performance of a team member, constant elasticity of substitution production functions exhibiting diminishing returns used in economics and consumer theory, and others. Sketch valuation functions are particularly valuable for finding approximate solutions to optimization problems such as best set selection and welfare maximization. They enable computationally efficient evaluation of approximate value oracle queries and provide an approximation guarantee for the underlying optimization problem.

翻译：我们考虑对集合估值函数进行草图构建的问题，该函数定义为独立随机物品价值的估值函数的期望。我们证明：对于满足弱齐次条件的单调次可加或次模估值函数，或在某些其他条件下，存在物品价值的离散化分布，其支撑集大小为$O(k\log(k))$，由此生成的草图估值函数对任意基数不超过$k$的集合的价值查询，均可达到常数因子近似。这种离散化分布可通过算法为每个物品的价值分布独立高效计算。我们的结果适用于应用场景中产生的多种估值函数，包括对应于团队成员最优表现的团队价值函数、经济学与消费理论中体现边际报酬递减的恒定替代弹性生产函数等。草图估值函数在解决最优集合选择与福利最大化等优化问题中尤为有价值，它不仅能实现近似价值预言查询的高效计算，还为底层优化问题提供了近似性能保证。