We consider the problem of sketching a stochastic valuation function, defined as the expectation of a valuation function of independent random item values. We show that for monotone subadditive or submodular valuation functions that satisfy 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 $k$. These discretized distributions can be efficiently computed by an algorithm for each item's value distribution separately. The obtained sketch results are of interest for various optimization problems such as best set selection and welfare maximization problems.

翻译：我们考虑了设计一个随机物品独立价值估值功能的预期值的问题。我们发现,对于满足微弱同质性条件或某些其他条件的单质子相加或亚质值函数,单质子相加或亚质值函数,存在以美元(k\log(k))美元(k)美元(k)美元)支持大小分别分配的物品价值,从而产生一个常量近似值的草图估值函数,对于一组基本价值小于或等于美元的项目的任何价值查询,这些分解的分布可以通过每种物品价值分布的算法分别有效计算。获得的草图结果对最佳选择和福利最大化问题等各种优化问题感兴趣。