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.

翻译：潘多拉魔盒是不确定性决策中的一个核心问题，能够建模多种现实生活场景。在该问题中，给定 $n$ 个盒子，每个盒子有固定的开启成本，且其内部未知价值服从已知分布，仅当支付开启成本后方可获知。我们的目标是设计一种开启盒子的策略，以最小化所选价值与支付开启成本之和。本文重新审视了数值分布存在相关性的潘多拉魔盒问题（该问题最初由 Chawla 等人研究，arXiv:1911.01632）。我们证明了独立情形下的最优算法（即魏茨曼法则）可直接适用于相关情形。事实上，与先前工作相比，我们的算法在显著简化操作的同时，实现了更优的近似保证。最后，我们展示了如何仅通过对相关价值分布进行采样来实现该法则。具体而言，我们发现当样本数量与盒子数量呈多项式关系时，即可保证算法的有效性。