Weitzman's Rule for Pandora's Box with Correlations

TL;DR

This paper demonstrates that Weitzman's rule for Pandora's Box remains optimal under correlated value distributions, offering improved approximation guarantees and a practical sampling-based implementation.

Contribution

It extends Weitzman's rule to correlated distributions, simplifying the approach and improving approximation guarantees compared to prior work.

Findings

Weitzman's rule applies directly to correlated cases.

The algorithm achieves better approximation guarantees.

A polynomial number of samples suffices for implementation.

Abstract

Pandora's Box is a central problem in decision making under uncertainty that can model various real life scenarios. In this problem we are given $n$ boxes, each with a fixed opening cost, and an unknown value drawn from a known distribution, only revealed if we pay the opening cost. Our goal is to find a strategy for opening boxes to minimize the sum of the value selected and the opening cost paid. In this work we revisit Pandora's Box when the value distributions are correlated, first studied in Chawla et al. (arXiv:1911.01632). We show that the optimal algorithm for the independent case, given by Weitzman's rule, directly works for the correlated case. In fact, our algorithm results in significantly improved approximation guarantees compared to the previous work, while also being substantially simpler. We finally show how to implement the rule given only sample access to the correlated distribution of values. Specifically, we find that a number of samples that is polynomial in the number of boxes is sufficient for the algorithm to work.