  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.
