Submodular Maximization using Test Scores

We consider the closely related problems of maximizing a submodular function subject to a cardinality constraint and that of maximizing the sum of submodular functions subject to a partition matroid contraint. Motivated by applications in team selection, we focus on settings where the function(s) can be expressed as the expected value of a symmetric submodular value function of a set of independent random variables with given prior distributions, and where, the algorithm designer may not be able to access the submodular function by means of value oracle queries. We consider a novel approach towards submodular maximization known as test score algorithms whose functioning is restricted to computing a test score for each element of the ground set and using these test scores to estimate the outputs of oracle calls. Our main contributions are test score algorithms that yield constant factor and logarithmic approximations respectively for the two problems along with new insights pertaining to function sketching. We also identify necessary and sufficient conditions under which test score algorithms provide constant-factor approximations for the submodular maximization problem. We evaluate the quality of our algorithms for team selection using data from a popular online labour platform for software development.