Submodular Maximization using Test Scores

Abstract: We study the canonical problem of maximizing a stochastic submodular function subject to a cardinality constraint, where the goal is to select a subset from a ground set of items with uncertain individual performances to maximize their expected group value. Although near-optimal algorithms have been proposed for this problem, practical concerns regarding scalability, compatibility with distributed implementation, and expensive oracle queries persist in large-scale applications. Motivated by online platforms that rely on individual item scores for content recommendation and team selection, we propose a special class of algorithms that select items based solely on individual performance measures known as test scores. The central contribution of this work is a novel and systematic framework for designing test score based algorithms for a broad class of naturally occurring utility functions. We introduce a new scoring mechanism that we refer to as replication test scores and prove that as long as the objective function satisfies a diminishing returns property, one can leverage these scores to compute solutions that are within a constant factor of the optimum. We then extend our results to the more general stochastic submodular welfare maximization problem, where the goal is to select items and assign them to multiple groups to maximize the sum of the expected group values. For this more difficult problem, we show that replication test scores can be used to develop an algorithm that approximates the optimum solution up to a logarithmic factor. The techniques presented in this work bridge the gap between the rigorous theoretical work on submodular optimization and simple, scalable heuristics that are useful in certain domains.