A Constructive Prophet Inequality Approach to The Adaptive ProbeMax Problem

In the adaptive ProbeMax problem, given a collection of mutually-independent random variables $X_1, \ldots, X_n$, our goal is to design an adaptive probing policy for sequentially sampling at most $k$ of these variables, with the objective of maximizing the expected maximum value sampled. In spite of its stylized formulation, this setting captures numerous technical hurdles inherent to stochastic optimization, related to both information structure and efficient computation. For these reasons, adaptive ProbeMax has served as a test bed for a multitude of algorithmic methods, and concurrently as a popular teaching tool in courses and tutorials dedicated to recent trends in optimization under uncertainty. The main contribution of this paper consists in proposing a novel method for upper-bounding the expected maximum reward of optimal adaptive probing policies, based on a simple min-max problem. Equipped with this method, we devise purely-combinatorial algorithms for deterministically computing feasible sets whose vicinity to the adaptive optimum is analyzed through prophet inequality ideas. Consequently, this approach allows us to establish improved constructive adaptivity gaps for the ProbeMax problem in its broadest form, where $X_1, \ldots, X_n$ are general random variables, making further advancements when $X_1, \ldots, X_n$ are continuous.