Abstract
In this work, we present an evolutionary omputation-based solution to the circle packing problem (ECPP). The circle packing problem consists of placing a set of circles into a larger containing circle without overlaps: a problem known to be NP-hard. Given the impossibility to solve this problem efficiently, traditional and heuristic methods have been proposed to solve it. A naïve representation for chromosomes in a population-based heuristic search leads to high probabilities of violation of the problem constraints, i.e., overlapping. To convert solutions that violate constraints into ones that do not (i.e., feasible solutions), in this paper we propose two repair mechanisms. The first one considers every circle as an elastic ring and overlaps create repulsion forces that lead the circles to positions where the overlaps are resolved. The second one forms a Delaunay triangulation with the circle centers and repairs the circles in each triangle at a time, making sure repaired triangles are not modified later on. Based on the proposed repair heuristics, we present the results of the solution to the CPP problem to a set of unit circle problems (whose exact optimal solutions are known). These benchmark problems are solved using genetic algorithms, evolutionary strategies, particle swarm optimization, and differential evolution. The performance of the solutions is compared to those known solutions based on the packing density. We then perform a series of experiments to determine the performance of ECPP with non-unitary circles. First, we compare ECPP’s results to those of a public competition, which stand as the world record for that particular instance of the non-unitary CPP. On a second set of experiments, we control the variance of the size of the circles. In all experiments, ECPP yields satisfactory near-optimal solutions.
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Although all experiments were also performed using genetic algorithms (GA), evolutionary strategies (ES), differential evolution (DE), and PSO, for the sake of brevity, only the GA and DE were included in the reports, since they were the ones that performed the best.
References
Addis B, Locatelli M, Schoen F (2008) Efficiently packing unequal disks in a circle. Oper Res Lett 36(1):37–42
Al-Modahka I, Hifi M, M’Hallah R (2011) Packing circles in the smallest circle: an adaptive hybrid algorithm. J Oper Res Soc 62(11):1917–1930
Castillo I, Kampas FJ, Pintér JD (2008) Solving circle packing problems by global optimization: numerical results and industrial applications. Eur J Oper Res 191(3):786–802
Deb K (2004) A population-based algorithm-generator for real-parameter optimization. Soft Comput 9:236–253
Demaine ED, Fekete SP, Lang RJ (2010) Circle packing for origami design is hard. arXiv:1008.1224
Dowsland KA, Gilbert M, Kendall G (2007) A local search approach to a circle cutting problem arising in the motor cycle industry. J Oper Res Soc 58(4):429–438
Francesco C, Cerrone C, Cerulli R (2014) A tabu search approach for the circle packing problem. In: 17th international conference on network-based information systems
Harary F, Randolph W, Mezey PG (1996) A study of maximum unit-circle caterpillars-tools for the study of the shape of adsorption patterns. Discret Appl Math 67(1–3):127–135
Hifi M, M’Hallah R (2009) Beam search and non-linear programming tools for the circular packing problem. Int J Math Oper Res 1(4):476–503
Lee D-T, Schachter BJ (1980) Two algorithms for constructing a delaunay triangulation. Int J Comput Inf Sci 9(3):219–242
Michalewicz Z (1996) Genetic algorithms+datastructures=evolutionary programs, 3rd edn. Springer, Berlin
Mladenovic N, Plastria F, Urosevic D (2005) Reformulation descent applied to circle packing problems. Comput Oper Res 32(9):2419–2434
Nordbakke MW, Ryum N, Hunderi O (2004) Curvilinear polygons, finite circle packings, and normal grain growth. Mater Sci Eng A 385(1–2):229–234
Pintér JD, Kampas FJ (2006) Nonlinear optimization in mathematica with math-optimizer professional. Math Educ Res 10:1–18
Shi Y-J, Liu Z-C, Ma S (2010) An improved evolution strategy for constrained circle packing problem. In: Advanced intelligent computing theories and applications. Springer, Berlin, pp 86–93
Specht E (1999) Packomania web site. http://www.packomania.com/
Sugihara K, Sawai M, Sano H, Kim DS, Kim D (2004) Disk packing for the estimation of the size of a wire bundle. Jpn J Ind Appl Math 21(3):259–278
Szabó PG, Markót MC, Csendes T, Specht E, Casado LG, García I (2006) New approaches to circle packing in a square. Springer, Berlin
Wang H, Huang W, Zhang Q, Xu D (2002) An improved algorithm for the packing of unequal circles within a larger containing circle. Eur J Oper Res 141(2):440–453
Wenqi H, Yan K (2004) A short note on a simple search heuristic for the diskspacking problem. Ann Oper Res 1–4:101–108
Xu Y-C, Xiao R-B, Amos M (2007) A novel genetic algorithm for the layout optimization problem. In: Evolutionary computation 2007 CEC 2007, pp 3938–3943
Yan-Jun S, Yi-Shou W, Long W, Hong-Fei T (2012) A layout pattern based particle swarm optimization for constrained packing problems. Inf Technol J 11:1722–1729
Zhang DF, Deng AS (2005) An effective hybrid algorithm for the problem of packing circles into a larger containing circle. Comput Oper Res 32(8):1941–1951
Zhi-Qin Q, Hong-Fei T, Zhi-Guo S (2001) Human–computer interactive genetic algorithm and its application to constrained layout optimization. Chin J Comput 5:553–560
Zimmermann A (2006) Al zimmermann’s programming contests. http://www.recmath.org/contest/CirclePacking/index.php
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Flores, J.J., Martínez, J. & Calderón, F. Evolutionary computation solutions to the circle packing problem. Soft Comput 20, 1521–1535 (2016). https://doi.org/10.1007/s00500-015-1603-y
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DOI: https://doi.org/10.1007/s00500-015-1603-y