Block Stacking, Airplane Refueling, and Robust Appointment Scheduling

TL;DR

This paper proves that the block stacking problem with varying block sizes is NP-hard, explores its connection to airplane refueling and scheduling problems, and provides approximation algorithms for practical solutions.

Contribution

It establishes the NP-hardness of the generalized block stacking problem and links it to real-world scheduling challenges, offering approximation algorithms.

Findings

Block stacking with variable blocks is NP-hard.

Connections between block stacking, airplane refueling, and scheduling problems are demonstrated.

Polynomial-time approximation schemes are developed for specific cases.

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.