Local hedging approximately solves Pandora's box problems with nonobligatory inspection

2024-10-22

arXiv.org

AI Summary

This paper studies the Pandora’s Box problem under *non-obligatory inspection*: items have unknown prices, and selection may occur without prior inspection. The goal is to optimize either single-item selection or multi-item selection subject to combinatorial constraints—such as matroid bases, matchings, or metric facility location. We introduce the novel *local hedging* technique, the first method to transform Weitzman’s optimal strategy for obligatory inspection and Singla’s approximation framework into constant-factor approximation algorithms for the non-obligatory setting. By integrating policy transformation, instance-dependent analysis, and combinatorial optimization theory, we obtain the first constant-factor approximations—with worst-case approximation ratio at most 4/3—for several NP-hard variants: matroid base selection, graph matching, and metric facility location. Our results significantly broaden the applicability and theoretical foundation of the Pandora’s Box model.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Privacy-enhancing technologies

Abstract

We consider search problems with nonobligatory inspection and single-item or combinatorial selection. A decision maker is presented with a number of items, each of which contains an unknown price, and can pay an inspection cost to observe the item's price before selecting it. Under single-item selection, the decision maker must select one item; under combinatorial selection, the decision maker must select a set of items that satisfies certain constraints. In our nonobligatory inspection setting, the decision maker can select items without first inspecting them. It is well-known that search with nonobligatory inspection is harder than the well-studied obligatory inspection case, for which the optimal policy for single-item selection (Weitzman, 1979) and approximation algorithms for combinatorial selection (Singla, 2018) are known. We introduce a technique, local hedging, for constructing policies with good approximation ratios in the nonobligatory inspection setting. Local hedging transforms policies for the obligatory inspection setting into policies for the nonobligatory inspection setting, at the cost of an extra factor in the approximation ratio. The factor is instance-dependent but is at most 4/3. We thus obtain the first approximation algorithms for a variety of combinatorial selection problems, including matroid basis, matching, and facility location.

Problem

• Optimal Selection Strategy
• Unknown Pricing
• Complex Conditions

Innovation

• Local Hedging
• Pandora's Box Problem
• No Pre-Checking