Pandora's Box Problem With Time Constraints

TL;DR

This paper introduces a comprehensive framework for the Pandora's Box problem incorporating time constraints, providing efficient approximation algorithms for complex variants where rewards, costs, and inspection times are time-dependent.

Contribution

It develops a general model capturing time effects in Pandora's Box problems and offers the first constant-factor approximation algorithms for these NP-hard variants.

Findings

Efficient constant-factor approximation for the general Pandora's Box Over Time problem.

Improved algorithms for special cases with no processing time or fixed time slots.

Demonstrates the problem's NP-hardness and provides practical approximation solutions.

Abstract

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).