Pandora's Box Problem with Order Constraints
Shant Boodaghians, Federico Fusco, Philip Lazos, Stefano Leonardi
TL;DR
This work generalizes Pandora's Box by incorporating order constraints, notably tree-like precedence, and shows that optimal search rules exist in the form of threshold strategies with per-box reservation values that account for future access via stopping times. It establishes polynomial-time computability of these thresholds on trees and unions of lines, and introduces the macrobox decomposition to handle depth-breadth tradeoffs. The paper also proves strong hardness results for broader constraint classes (matroids and DAGs), while offering approximation via adaptivity gaps and learning-based approaches, including sample complexity bounds for additive ε-approximations. These results advance the practical deployment of adaptive search under structured precedence constraints and guide future work on more general constraint families and data-driven learning of distributions.
Abstract
The Pandora's Box Problem, originally formalized by Weitzman in 1979, models selection from set of random, alternative options, when evaluation is costly. This includes, for example, the problem of hiring a skilled worker, where only one hire can be made, but the evaluation of each candidate is an expensive procedure. Weitzman showed that the Pandora's Box Problem admits an elegant, simple solution, where the options are considered in decreasing order of reservation value,i.e., the value that reduces to zero the expected marginal gain for opening the box. We study for the first time this problem when order - or precedence - constraints are imposed between the boxes. We show that, despite the difficulty of defining reservation values for the boxes which take into account both in-depth and in-breath exploration of the various options, greedy optimal strategies exist and can be efficiently computed for tree-like order constraints. We also prove that finding approximately optimal adaptive search strategies is NP-hard when certain matroid constraints are used to further restrict the set of boxes which may be opened, or when the order constraints are given as reachability constraints on a DAG. We complement the above result by giving approximate adaptive search strategies based on a connection between optimal adaptive strategies and non-adaptive strategies with bounded adaptivity gap for a carefully relaxed version of the problem.