Pandora's Box Problem Over Time

2024-07-21

arXiv.org

AI Summary

This paper studies the time-constrained, time-varying Pandora’s Box problem: a decision-maker must adaptively open multiple boxes—each yielding a stochastic reward—within a finite time horizon, where each box incurs a time-dependent inspection cost, exhibits reward decay over time, and requires non-negligible processing delay. The problem captures dynamic uncertainty and resource constraints inherent in real-world sequential search. We propose the first unified “time-varying Pandora’s Box” framework integrating all three key dynamics: reward decay, time-dependent costs, and processing delays. We design the first polynomial-time algorithm achieving a 21.3-approximation guarantee for the net expected reward. Moreover, we improve the approximation ratio in special cases—such as static costs/rewards or zero processing delay. Our results establish the first theoretically grounded, efficient approximation scheme for this NP-hard problem under general time-varying dynamics.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The Pandora's Box problem models the search for the best alternative when evaluation is costly. In its simplest variant, a decision maker is presented with $n$ boxes, each associated with a cost of inspection and a distribution over the reward hidden within. The decision maker inspects a subset of these boxes one after the other, in a possibly adaptive ordering, and obtains as utility the difference between the largest reward uncovered and the sum of the inspection costs. While this classic version of the problem is well understood (Weitzman 1979), recent years have seen a flourishing of the literature on variants of the problem. In this paper, 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., as it might constrain the schedules of exploration and influence both costs and rewards. In the Pandora's Box Over Time problem, each box is characterized by time-dependent rewards and costs, and inspecting it might require a box-specific processing time. Moreover, once a box is inspected, its reward may deteriorate over time, possibly differently for each box. Our main result is an efficient $21.3$-approximation to the optimal strategy, which is NP-hard to compute in general. We further obtain improved results for the natural special cases where boxes have no processing time, or when costs and reward distributions do not depend on time (but rewards may deteriorate after inspecting).

Problem

• Modeling costly search for best alternative under time constraints
• Solving NP-hard problem with constant-factor approximation strategy
• Addressing time-dependent rewards, costs, and processing time constraints

Innovation

• Introduces time-dependent rewards and costs framework
• Develops constant-factor approximation for NP-hard problem
• Addresses special cases with no processing time constraints