Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Block Stacking, Airplane Refueling, and Robust Appointment Scheduling

Simon Gmeiner, Andreas S. Schulz

TL;DR

The paper analyzes the Block-Stacking Problem with blocks of varying width and mass, proving NP-hardness and revealing deep connections to the Airplane Refueling Problem and Robust Appointment Scheduling. It develops structural results for optimal stacks, showing a two-part composition of counterweights and right-aligned blocks and deriving a concrete overhang formula. By relating no-counterbalancing BSP to Airplane Refueling, it leverages a PTAS to obtain a $(1+\varepsilon)$-approximation in that variant and a $(2+\varepsilon)$-approximation for the general case. The work highlights how a classical puzzle informs practical scheduling under uncertainty through cross-problem algorithmic transfers, while leaving open questions about exact polynomial-time solutions for the general problem.

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 $(1+ε)$-approximation algorithm, for Block Stacking without counterbalancing and a $(2+ε)$-approximation algorithm for the general case.

Block Stacking, Airplane Refueling, and Robust Appointment Scheduling

TL;DR

The paper analyzes the Block-Stacking Problem with blocks of varying width and mass, proving NP-hardness and revealing deep connections to the Airplane Refueling Problem and Robust Appointment Scheduling. It develops structural results for optimal stacks, showing a two-part composition of counterweights and right-aligned blocks and deriving a concrete overhang formula. By relating no-counterbalancing BSP to Airplane Refueling, it leverages a PTAS to obtain a -approximation in that variant and a -approximation for the general case. The work highlights how a classical puzzle informs practical scheduling under uncertainty through cross-problem algorithmic transfers, while leaving open questions about exact polynomial-time solutions for the general problem.

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 -approximation algorithm, for Block Stacking without counterbalancing and a -approximation algorithm for the general case.
Paper Structure (11 sections, 17 theorems, 38 equations, 10 figures)

This paper contains 11 sections, 17 theorems, 38 equations, 10 figures.

Table of Contents

  1. Introduction.
  2. The Block-Stacking Problem.
  3. Stacking Configurations.
  4. Simple Stacking Rules.
  5. NP-Hardness.
  6. Solving the Partition Problem via the Block-Stacking Problem.
  7. Block Stacking without Counterbalancing.
  8. The Airplane Refueling Problem.
  9. Robust Appointment Scheduling.
  10. A $\boldsymbol{(2+\epsilon)}$-Approximation for Block Stacking.
  11. Conclusion.

Key Result

Theorem 1

A stack of maximum overhang with protruding block $p \in \{1,\dots,n\}$ satisfies the following two conditions:

Figures (10)

  • Figure 1: The harmonic stack and the alternative balanced stack configuration with maximum overhang for identical blocks of width two.
  • Figure 2: Without counterbalancing, blocks of the same height, ordered from top to bottom by non-increasing width, achieve the maximum possible overhang. Using some of the blocks as counterweights allows for an even larger overhang.
  • Figure 3: Optimal stack configurations with and without counterbalancing. In this and all subsequent figures, we assume that the mass of a block is proportional to the product of its width and height.
  • Figure 4: Both stacks are balanced, but only the one on the right is right-aligned at block $i$. The indicated adjustments for the left stack as per Theorem \ref{['result:theorem:stack_configuration']} lead to a larger overhang.
  • Figure 5: The general structure of a stack with maximum overhang. The top three blocks serve as counterweights and have no direct contribution to the overhang. The right-aligned blocks incrementally form the overhang. Due to counterweights, the protruding block can contribute to the overhang with more than half its width.
  • ...and 5 more figures

Theorems & Definitions (30)

  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Remark
  • Corollary 3
  • proof
  • Lemma 4
  • proof
  • Definition : Partition Problem
  • ...and 20 more