A prophet inequality based approach to the adaptive ProbeTop$$\varvec{K}$$ problem

Abstract In the adaptive ProbeTop $$K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> problem, given a collection of mutually independent random variables $$X_1, \ldots , X_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , our goal is to design an adaptive probing policy to sample these variables in a sequence of T stages, with the objective of maximizing the expected sum of the K highest rewards 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, special cases and variants of this problem have 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 reward of optimal adaptive probing policies, based on a simple Min-Max problem. Equipped with this method, we devise a 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 ProbeTop $$K$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> problem in its broadest form, where $$X_1, \ldots , X_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> are general random variables, making further advancements when $$X_1, \ldots , X_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> are continuous.