The Pandora's Box problem models the search for the best alternative when evaluation is costly. In the simplest variant, a decision maker is presented with $n$ boxes, each associated with a cost of inspection and a hidden random reward. The decision maker inspects a subset of these boxes one after the other, in a possibly adaptive order, and gains the difference between the largest revealed reward and the sum of the inspection costs. Although this classic version is well understood (Weitzman 1979), there is a flourishing recent literature on variants of the problem. Here we introduce a general framework -- the Pandora's Box Over Time problem -- that captures a wide range of variants where time plays a role, e.g., by constraining the schedules of exploration and influencing costs and rewards. In our framework, boxes have time-dependent rewards and costs, whereas inspection may require a box-specific processing time. Moreover, once a box is inspected, its reward may deteriorate over time. Our main result is an efficient constant-factor approximation to the optimal strategy for the Pandora's Box Over Time problem, which is generally NP-hard to compute. We further obtain improved results for the natural special cases where boxes have no processing time, boxes are available only in specific time slots, or when costs and reward distributions are time-independent (but rewards may still deteriorate after inspection).

翻译：潘多拉魔盒问题为评估成本高昂时的最优替代方案搜索建立了模型。在最简单的变体中，决策者面对 n 个盒子，每个盒子关联着一定的检查成本和隐藏的随机奖励。决策者按可能自适应的顺序依次检查这些盒子的一个子集，最终获得的收益等于已揭示的最大奖励与检查总成本之差。尽管这个经典版本已被充分理解（Weitzman 1979），但近期关于该问题变体的研究文献正蓬勃发展。本文提出了一个通用框架——时变潘多拉魔盒问题——该框架涵盖了时间因素起作用的多种变体，例如通过约束探索调度并影响成本与奖励。在我们的框架中，盒子具有随时间变化的奖励和成本，而检查过程可能需要盒子特定的处理时间。此外，盒子被检查后，其奖励可能随时间衰减。我们的主要成果是为时变潘多拉魔盒问题的最优策略提出了一个高效的常数因子近似解，而该问题在一般情况下是 NP 难计算的。我们进一步针对以下自然特例获得了改进结果：盒子无需处理时间、盒子仅在特定时间窗可用、或成本与奖励分布与时间无关（但检查后奖励仍可能衰减）的情形。