# 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}
}
• Published 1 November 2002
• Computer Science, Mathematics
• Algorithmica
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).

### Heterogeneous matrix-matrix multiplication or partitioning a square into rectangles: NP-completeness and approximation algorithms

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Proceedings Ninth Euromicro Workshop on Parallel and Distributed Processing
• 2001
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

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### On three soft rectangle packing problems with guillotine constraints

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J. Glob. Optim.
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It is shown that the first problem can be solved to optimality in $${{\mathcal {O}}}(n \log n)$$O(nlogn), while the two others are NP-hard.

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• Computer Science
2016 IEEE International Parallel and Distributed Processing Symposium (IPDPS)
• 2016
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.

### Cuboid Partitioning for Parallel Matrix Multiplication on Heterogeneous Platforms

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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.

### Recent Advances in Matrix Partitioning for Parallel Computing on Heterogeneous Platforms

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IEEE Transactions on Parallel and Distributed Systems
• 2019
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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