The Minimum Expectation Selection Problem

TL;DR

This paper investigates the computational complexity of selecting distributions to optimize the expected minimum value, revealing NP-completeness and approximation schemes depending on the number of support points.

Contribution

It characterizes the complexity of min-min and max-min expectation selection problems, providing NP-hardness results and polynomial algorithms for fixed support sizes.

Findings

NP-complete for fixed support size greater than 2

Fully polynomial approximation scheme exists for certain cases

Polynomial solvability for max-min problem with fixed support size

Abstract

We define the min-min expectation selection problem (resp. max-min expectation selection problem) to be that of selecting k out of n given discrete probability distributions, to minimize (resp. maximize) the expectation of the minimum value resulting when independent random variables are drawn from the selected distributions. We assume each distribution has finitely many atoms. Let d be the number of distinct values in the support of the distributions. We show that if d is a constant greater than 2, the min-min expectation problem is NP-complete but admits a fully polynomial time approximation scheme. For d an arbitrary integer, it is NP-hard to approximate the min-min expectation problem with any constant approximation factor. The max-min expectation problem is polynomially solvable for constant d; we leave open its complexity for variable d. We also show similar results for binary selection problems in which we must choose one distribution from each of n pairs of distributions.