Sketching stochastic valuation functions

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.