On bounded Wang tilings

TL;DR

This paper explores bounded Wang tilings, introducing four integer programming formulations and a heuristic algorithm to generate valid tilings, with applications in materials engineering and insights into known aperiodic tile sets.

Contribution

It presents novel integer programming models and a heuristic for bounded Wang tilings, along with analysis of tile set properties and performance benchmarking.

Findings

Heuristic algorithms achieve competitive results quickly.

Identified errors in two well-known aperiodic tile sets.

Provided approximation guarantees for the maximum-cover problem.

Abstract

Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging applications in materials engineering, we consider the bounded version of the tiling problem and offer four integer programming formulations to construct valid or nearly-valid Wang tilings: a decision, maximum-rectangular tiling, maximum cover, and maximum adjacency constraint satisfaction formulations. To facilitate a finer control over the resulting tilings, we extend these programs with tile-based, color-based, packing, and variable-sized periodic constraints. Furthermore, we introduce an efficient heuristic algorithm for the maximum-cover variant based on the shortest path search in directed acyclic graphs and derive simple modifications to provide a $1/2$ approximation guarantee for arbitrary tile sets, and a $2/3$ guarantee for tile sets with cyclic transducers. Finally, we benchmark the performance of the integer programming formulations and of the heuristic algorithms showing that the heuristics provides very competitive outputs in a fraction of time. As a by-product, we reveal errors in two well-known aperiodic tile sets: the Knuth tile set contains a tile unusable in two-way infinite tilings, and the Lagae corner tile set is not aperiodic.