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The Pandora's Box Problem, originally formalized by Weitzman in 1979, modelsselection from set of random, alternative options, when evaluation is costly.This includes, for example, the problem of hiring a skilled worker, where onlyone hire can be made, but the evaluation of each candidate is an expensiveprocedure. Weitzman showed that the Pandora's Box Problem admits an elegant,simple solution, where the options are considered in decreasing order ofreservation value,i.e., the value that reduces to zero the expected marginalgain 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 takeinto account both in-depth and in-breath exploration of the various options,greedy optimal strategies exist and can be efficiently computed for tree-likeorder constraints. We also prove that finding approximately optimal adaptivesearch strategies is NP-hard when certain matroid constraints are used tofurther restrict the set of boxes which may be opened, or when the orderconstraints are given as reachability constraints on a DAG. We complement theabove result by giving approximate adaptive search strategies based on aconnection between optimal adaptive strategies and non-adaptive strategies withbounded adaptivity gap for a carefully relaxed version of the problem.

Pandora's Box Problem with Order Constraints