Annals of Operations Research

, Volume 271, Issue 2, pp 831–851

# Minimum tiling of a rectangle by squares

• Michele Monaci
• André Gustavo dos Santos
Original Research

## Abstract

We consider a two-dimensional problem in which one is required to split a given rectangular bin into the smallest number of items. The resulting items must be squares to be packed, without overlapping, into the bin so as to cover all the given rectangle. We present a mathematical model and a heuristic algorithm that is proved to find the optimal solution in some special cases. Then, we introduce a relaxation of the problem and present different exact approaches based on this relaxation. Finally, we report computational experiments on the performances of the algorithms on a large set of randomly generated instances.

## Keywords

Two-dimensional packing Mathematical models Exact algorithms Computational experiments

## Notes

### Acknowledgements

This research was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil under Grant PVE 030479/2013-01. Thanks are due to an anonymous referee for helpful comments.

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