# Partitioning a Square into Rectangles: NP-Completeness and Approximation Algorithms

@article{Beaumont2002PartitioningAS, title={Partitioning a Square into Rectangles: NP-Completeness and Approximation Algorithms }, author={Olivier Beaumont and Vincent Boudet and Fabrice Rastello and Yves Robert}, journal={Algorithmica}, year={2002}, volume={34}, pages={217-239} }

AbstractIn this paper we deal with two geometric problems arising from heterogeneous parallel computing: how to partition the unit square into p rectangles of given area s1, s2, . . . ,sp (such that Σi=1p si = 1 ), so as to minimize either (i) the sum of the p perimeters of the rectangles or (ii) the largest perimeter of the p rectangles? For both problems, we prove NP-completeness and we introduce a 7/4 -approximation algorithm for (i) and a $(2/\sqrt{3})$ -approximation algorithm for (ii).

## 45 Citations

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This paper proves NP-completeness and introduces approximation algorithms for partitioning the unit square into p rectangles so as to minimize either the sum of the p perimeters of the rectangles or the largest perimeter of thep rectangles.

### Minimum tiling of a rectangle by squares

- Computer ScienceAnn. Oper. Res.
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A mathematical model and a heuristic algorithm that is proved to find the optimal solution in some special cases of a two-dimensional problem in which one is required to split a given rectangular bin into the smallest number of items.

### Exact and approximation algorithms for a soft rectangle packing problem

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A polynomial time approximation algorithm is presented and an upper bound estimation on its approximation ratio is derived, which shows that adding the solutions from the approximation algorithm as advanced starter helps to reduce the overall solution time for proven global optimality.

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This paper considers the problem of partitioning a square into aset of zones of prescribed areas, while minimizing the overall size of their projections onto horizontal and vertical axes, and proposes a non-rectangular recursive partitioning (NRRP), whoseroximation ratio is 2/√3 ≃ 1.15.

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It is proved that, if a(R) ≥ Σ1≤i≤n a(ri) + 0.10103amax and amax ≤ 3(min{h(R), w( R)})2 hold for a amax = max1≡i≡n a (ri), then these n soft rectangles can be packed inside R so that the apect ratio of each rectangle ri is at most 3.

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This paper presents recent approaches that relax the restriction that all partitions be rectangles and uses the first exact approach to analyse how close to the known optimal solutions the NRRP algorithm is for small numbers of partitions.

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