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

TL;DR

This paper introduces local hedging, a technique that transforms policies from obligatory to nonobligatory inspection settings, providing the first approximation algorithms for complex search problems like matroid basis, matching, and facility location.

Contribution

The paper presents local hedging, a novel method that achieves near-optimal approximation ratios for nonobligatory inspection problems by leveraging existing policies.

Findings

Local hedging achieves an approximation factor of at most 4/3.

First approximation algorithms for nonobligatory inspection in combinatorial problems.

Applicable to problems like matroid basis, matching, and facility location.

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.