Weitzman's Rule for Pandora's Box with Correlations

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. [2020]. 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 also 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. This problem can be seen as being part of the "price of information" literature [Charikar et al., 2000, Gupta and Kumar, 2001, Chen et al., 2015b,a], where we can remove part of the uncertainty of the problem at hand by paying a price. In this line of work, more recent papers study the structure of approximately optimal rules for combinatorial problems [