Set Selection under Explorable Stochastic Uncertainty via Covering Techniques

TL;DR

This paper introduces algorithms for selecting minimal-value subsets under stochastic uncertainty, leveraging a novel connection to covering problems, and achieves near-optimal bounds in expectation, advancing the understanding of explorable uncertainty.

Contribution

The paper presents the first non-trivial stochastic algorithms for set selection under uncertainty, utilizing a new LP-based connection to covering problems to improve bounds.

Findings

Algorithms outperform adversarial bounds in expectation

Established a structural link to covering problems with uncertainty

Achieved nearly tight competitive ratio bounds

Abstract

Given subsets of uncertain values, we study the problem of identifying the subset of minimum total value (sum of the uncertain values) by querying as few values as possible. This set selection problem falls into the field of explorable uncertainty and is of intrinsic importance therein as it implies strong adversarial lower bounds for a wide range of interesting combinatorial problems such as knapsack and matchings. We consider a stochastic problem variant and give algorithms that, in expectation, improve upon these adversarial lower bounds. The key to our results is to prove a strong structural connection to a seemingly unrelated covering problem with uncertainty in the constraints via a linear programming formulation. We exploit this connection to derive an algorithmic framework that can be used to solve both problems under uncertainty, obtaining nearly tight bounds on the competitive ratio. This is the first non-trivial stochastic result concerning the sum of unknown values without further structure known for the set. With our novel methods, we lay the foundations for solving more general problems in the area of explorable uncertainty.