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On Guillotine Cutting Sequences

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Fidaa Abed, Parinya Chalermsook, José Correa, Andreas Karrenbauer, Pablo Pérez-Lantero, José A. Soto, and Andreas Wiese. On Guillotine Cutting Sequences. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 1-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.1

Abstract

Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack.
Keywords
• Guillotine cuts
• Rectangles
• Squares
• Independent Sets
• Packing

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References

1. Anna Adamaszek and Andreas Wiese. Approximation schemes for maximum weight independent set of rectangles. In Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2013), pages 400-409. IEEE, 2013.
2. Anna Adamaszek and Andreas Wiese. A quasi-PTAS for the two-dimensional geometric knapsack problem. In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), pages 1491-1505, 2015.
3. Nikhil Bansal, Alberto Caprara, Klaus Jansen, Lars Prädel, and Maxim Sviridenko. A structural lemma in 2-dimensional packing, and its implications on approximability. In Algorithms and Computation (ISAAC 2009), volume 5878 of LNCS, pages 77-86. Springer, 2009.
4. Parinya Chalermsook and Julia Chuzhoy. Maximum independent set of rectangles. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pages 892-901. SIAM, 2009.
5. Timothy M. Chan and Sariel Har-Peled. Approximation algorithms for maximum independent set of pseudo-disks. Discrete & Computational Geometry, 48:373-392, 2012.
6. Edward G Coffman, Jr, Michael R Garey, David S Johnson, and Robert Endre Tarjan. Performance bounds for level-oriented two-dimensional packing algorithms. SIAM J. Comput., 9(4):808-826, 1980.
7. Thomas Erlebach, Klaus Jansen, and Eike Seidel. Polynomial-time approximation schemes for geometric intersection graphs. SIAM Journal on Computing, 34(6):1302-1323, 2005.
8. Wenceslas Fernandez de la Vega and George S. Lueker. Bin packing can be solved within 1 + ε in linear time. Combinatorica, 1:349-355, 1981.
9. Aleksei V. Fishkin, Olga Gerber, Klaus Jansen, and Roberto Solis-Oba. Packing weighted rectangles into a square. In Mathematical Foundations of Computer Science (MFCS 2005), volume 2337 of LNCS, pages 352-363. Springer, 2005.
10. Rolf Harren. Approximating the orthogonal knapsack problem for hypercubes. In Automata, Languages and Programming (ICALP 2006), volume 4051 of LNCS, pages 238-249. Springer, 2006.
11. Klaus Jansen and Roberto Solis-Oba. New approximability results for 2-dimensional packing problems. In Mathematical Foundations of Computer Science (MFCS 2007), volume 4708 of LNCS, pages 103-114. Springer, 2007.
12. Klaus Jansen and Roberto Solis-Oba. A polynomial time approximation scheme for the square packing problem. In Proceedings of the 13th Integer Programming and Combinatorial Optimization Conference (IPCO 2008), pages 184-198. Springer, 2008.
13. Klaus Jansen and Guochuan Zhang. Maximizing the number of packed rectangles. In Proceedings of the 9th Scandinavian Workshop on Algorithm Theory (SWAT 2004), pages 362-371. Springer, 2004.
14. Klaus Jansen and Guochuan Zhang. Maximizing the total profit of rectangles packed into a rectangle. Algorithmica, 47(3):323-342, 2007.
15. János Pach and Gábor Tardos. Cutting glass. In Proceedings of the 16th Annual Symposium on Computational Geometry (SOCG 2000), pages 360-369. ACM, 2000.
16. Jorge Urrutia. Problem presented at ACCOTA'96: Combinatorial and Computational Aspects of Optimization, Topology, and Algebra, Taxco, Mexico, 1996.