Block Stacking, Airplane Refueling, and Robust Appointment Scheduling

2026-02-11

AI Summary

This study addresses the generalized block stacking problem—how to efficiently stack blocks of varying widths and masses to achieve maximum overhang. It establishes, for the first time, that this problem is NP-hard and demonstrates its equivalence to the airplane refueling problem and the robust appointment scheduling problem. Leveraging these connections, the authors devise two approximation algorithms: a (1+ε)-approximation for the weightless variant and a (2+ε)-approximation for the general case. This work not only constructs a theoretical bridge among three classical optimization problems but also partially resolves the long-standing open challenge of overhang maximization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Applications
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

How can a stack of identical blocks be arranged to extend beyond the edge of a table as far as possible? We consider a generalization of this classic puzzle to blocks that differ in width and mass. Despite the seemingly simple premise, we demonstrate that it is unlikely that one can efficiently determine a stack configuration of maximum overhang. Formally, we prove that the Block-Stacking Problem is NP-hard, partially answering an open question from the literature. Furthermore, we demonstrate that the restriction to stacks without counterweights has a surprising connection to the Airplane Refueling Problem, another famous puzzle, and to Robust Appointment Scheduling, a problem of practical relevance. In addition to revealing a remarkable relation to the real-world challenge of devising schedules under uncertainty, their equivalence unveils a polynomial-time approximation scheme, that is, a $(1+\epsilon)$-approximation algorithm, for Block Stacking without counterbalancing and a $(2+\epsilon)$-approximation algorithm for the general case.

Problem

• Block Stacking
• Airplane Refueling
• Robust Appointment Scheduling
• NP-hard
• Overhang Maximization

Innovation

• Block Stacking
• NP-hardness
• Approximation Algorithms
• Airplane Refueling Problem
• Robust Appointment Scheduling