Local hedging approximately solves Pandora's box problems with nonobligatory inspection
Ziv Scully, Laura Doval
TL;DR
This work introduces local hedging, a randomized committing policy for Pandora's box problems with nonobligatory inspection, preserving the simplicity and compositionality of Weitzman-style reservation policies. By assigning an item-specific hedging probability $p_n$ and mixing obligatory-inspection actions with inspection-free options, the method achieves a per-item local $\alpha$-approximation with $\alpha\le 4/3$ and extends to combinatorial selection via composition with greedy (or Frugal) obligatory-inspection algorithms, yielding a $\frac{4}{3}\beta$-approximation when the underlying greedy is a $\beta$-approximation. The approach relies on surrogate prices ($W^{\mathsf{OI}}$, $W^{\mathsf{NI}}$, and $W^{\mathsf{LH}}$) and a local approximation framework that bounds the optimal NI cost from below by $\mathbb{E}[\min_n W^{\mathsf{NI}}_n]$ (and the OI cost by $\mathbb{E}[\min_n W^{\mathsf{OI}}_n]$ in the obligatory case). The results provide the first approximation algorithms for a broad class of combinatorial Pandora's box problems under nonobligatory inspection (e.g., matroid basis, matching, facility location), with explicit construction and tightness discussions, and they suggest extensions to reward-maximization settings and Markovian bandit superprocesses. Overall, local hedging offers a simple, scalable, and compositional toolkit for tackling nonobligatory inspection in both single-item and combinatorial Pandora's box problems, delivering practical approximation guarantees where exact solutions are intractable.
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.