Block Stacking, Airplane Refueling, and Robust Appointment Scheduling

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.

💡 Research Summary

The paper studies a generalized version of the classic block‑stacking puzzle in which each block may have its own width and mass. The objective is to arrange the blocks into a vertically stacked structure that extends as far as possible beyond the edge of a table while remaining physically balanced. The authors first formalize the physics: for a stack of n blocks, let w_i be half‑width, m_i the mass, and x_i the horizontal coordinate of the block’s centre. The stack is balanced if, for every i = 1,…,n‑1, the centre of gravity of the top i blocks lies within the support interval of block i + 1. Assuming the overall centre of gravity is exactly at the table edge, the overhang is defined as max_i (x_i + w_i).

A key structural result (Theorem 1) shows that any optimal stack can be decomposed into two parts: a “protruding block” p and the blocks above it, which must all be right‑aligned (the centre of gravity of the blocks above each block coincides with the right edge of the block below). The blocks below p act only as counterweights; they contribute to the overhang indirectly by increasing the total mass above p. Using this decomposition the overhang can be expressed analytically as

 O = w_p·(2 − m_p/M_p) + ∑_{i>p} w_i·m_i/M_i,

where M_i denotes the cumulative mass of the top i blocks. This formula reveals that a heavier counterweight below p can increase the contribution of p, while heavier blocks above a right‑aligned block reduce its marginal contribution.

From this insight the authors derive simple ordering rules. Lemma 2 proves that for any two adjacent right‑aligned blocks a (above) and b (below) with the same set of blocks above them, the inequality w_a·M + m_a ≥ w_b·M + m_b must hold, where M is the total mass above a. This formalizes the intuitive guideline “place wider and lighter blocks higher”.

The paper then proves that the Block‑Stacking Problem is NP‑hard. By constructing blocks whose widths and masses encode a 0‑1 knapsack (or partition) instance, the decision version “does there exist a stack with overhang at least K?” is shown to be equivalent to the knapsack decision problem. This settles, in part, an open conjecture by Treibey that the problem should be computationally intractable when blocks have heterogeneous dimensions.

A striking contribution is the identification of an equivalence between three seemingly unrelated problems: (i) block stacking without counterweights, (ii) the Airplane Refueling Problem, and (iii) Robust Appointment Scheduling. In the airplane refueling model, aircraft share fuel in flight; the amount of fuel carried corresponds to block mass, and fuel consumption rate corresponds to block width. Maximizing the total flight distance is mathematically identical to maximizing overhang when no counterweights are allowed. In robust appointment scheduling, jobs have uncertain processing times; the goal is to minimize the worst‑case lateness. By mapping job lengths and uncertainty intervals to block widths and masses, the scheduling objective again matches the overhang objective.

Because both the airplane refueling problem and robust appointment scheduling admit polynomial‑time approximation schemes (PTAS) that achieve a (1 + ε) factor for any fixed ε > 0, the same PTAS can be directly transferred to the block‑stacking problem without counterweights. Consequently the authors obtain a (1 + ε)‑approximation algorithm for that variant. When counterweights are permitted, the structural analysis introduces an additional constant factor of 2, leading to a (2 + ε)‑approximation algorithm. Both algorithms run in time polynomial in the number of blocks (though exponential in 1/ε, as typical for PTAS).

The paper concludes by discussing practical implications. The equivalence shows that techniques developed for fuel‑sharing in aerospace or for robust scheduling in operations management can be repurposed for a purely geometric puzzle, and vice‑versa. It also highlights that seemingly simple physical puzzles can encode hard combinatorial optimisation problems. Future work suggested includes tightening the approximation factor for the general case, extending the model to three‑dimensional blocks (including depth and thickness), and incorporating dynamic or stochastic elements into the stacking process. Overall, the work bridges physics‑based puzzles, computational complexity, and algorithmic optimisation, delivering both theoretical insight and practical approximation tools.