Abstract: Weitzman [1979] introduced Pandora's box problem as a mathematical model of sequential search with inspection costs, in which a searcher is allowed to select a prize from one of n alternatives. Several decades later, Doval [2018] introduced a close version of the problem, where the searcher does not need to incur the inspection cost of an alternative, and can select it uninspected. Unlike the original problem, the optimal solution to the nonobligatory inspection variant is proved to need adaptivity [Doval 2018], and by recent work of Fu, Li, and Liu [2022], finding the optimal solution is NP-hard.
Our first main result is a structural characterization of the optimal policy: We show there exists an optimal policy that follows only two different pre-determined orders of inspection, and transitions from one to the other at most once. Our second main result is a polynomial time approximation scheme (PTAS). Our proof involves a novel reduction to a framework developed by Fu, Li, and Xu [2018], utilizing our optimal two-phase structure. Furthermore, we show Pandora's problem with nonobligatory inspection belongs to class NP, which by using the hardness result of Fu, Li, and Liu [2022], settles the computational complexity class of the problem. Finally, we provide a tight 0.8 approximation and a novel proof for committing policies [Beyhaghi and Klinberg 2019] (informally, the set of nonadaptive policies) for general classes of distributions, which was previously shown only for discrete and finite distributions [Guha, Munagala, and Sarkar 2008].
Based on joint work with Robert Kleinberg (EC 2019) and Linda Cai (STOC 2023).