Pandora Box Problem with Nonobligatory Inspection: Hardness and Approximation Scheme

Weitzman (1979) introduced the Pandora Box problem as a model for sequential search with inspection costs, and gave an elegant index-based policy that attains provably optimal expected payoff. In various scenarios, the searching agent may select an option without making a costly inspection. The variant of the Pandora box problem with non-obligatory inspection has attracted interest from both economics and algorithms researchers. Various simple algorithms have proved suboptimal, with the best known 0.8-approximation algorithm due to Guha et al. (2008). No hardness result for the problem was known. In this work, we show that it is NP-hard to compute an optimal policy for Pandora's problem with nonobligatory inspection. We also give a polynomial-time approximation scheme (PTAS) that computes policies with an expected payoff at least $(1 - ε)$-fraction of the optimal, for arbitrarily small $ε> 0$. On the side, we show the decision version of the problem to be in NP.