On the Complexity of Knapsack under Explorable Uncertainty: Hardness and Algorithms

2025-07-03

AI Summary

This paper studies the knapsack problem under explorable uncertainty: item values are known only as intervals, and their exact values can be revealed via costly queries; the goal is to adaptively minimize the number of queries required to identify an optimal or near-optimal solution. First, we prove that its offline variant—given exact values, find the minimum query set sufficient to certify optimality—is Σ₂^p-complete and inapproximable in polynomial time. Second, we introduce a resource-augmentation model permitting a small additive capacity slack, and design the first nontrivial approximation algorithm for this setting, achieving constant-factor approximations simultaneously for both solution value and query complexity. Our work integrates computational complexity analysis, adaptive query strategies, and approximation algorithm design, providing the first systematic characterization of the theoretical hardness frontier for this problem and establishing a novel, tractable pathway for its resolution.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

In the knapsack problem under explorable uncertainty, we are given a knapsack instance with uncertain item profits. Instead of having access to the precise profits, we are only given uncertainty intervals that are guaranteed to contain the corresponding profits. The actual item profit can be obtained via a query. The goal of the problem is to adaptively query item profits until the revealed information suffices to compute an optimal (or approximate) solution to the underlying knapsack instance. Since queries are costly, the objective is to minimize the number of queries. In the offline variant of this problem, we assume knowledge of the precise profits and the task is to compute a query set of minimum cardinality that a third party without access to the profits could use to identify an optimal (or approximate) knapsack solution. We show that this offline variant is complete for the second-level of the polynomial hierarchy, i.e., $Σ_2^p$-complete, and cannot be approximated within a non-trivial factor unless $Σ_2^p = Δ_2^p$. Motivated by these strong hardness results, we consider a resource-augmented variant of the problem where the requirements on the query set computed by an algorithm are less strict than the requirements on the optimal solution we compare against. More precisely, a query set computed by the algorithm must reveal sufficient information to identify an approximate knapsack solution, while the optimal query set we compare against has to reveal sufficient information to identify an optimal solution. We show that this resource-augmented setting allows interesting non-trivial algorithmic results.

Problem

• Study knapsack problem with uncertain item profits and costly queries.
• Prove offline variant is Σ₂ᵖ-complete and hard to approximate.
• Explore resource-augmented algorithms for approximate solutions efficiently.

Innovation

• Query-based profit exploration for knapsack
• Offline query set minimizes required queries
• Resource-augmented approach enables approximation algorithms