# The Problem of the Optimal Packing of the Equal Circles for Special Non-Euclidean Metric

Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 661)

## Abstract

The optimal packing problem of equal circles (2-D spheres) in a bounded set P in a two-dimensional metric space is considered. The sphere packing problem is to find an arrangement in which the spheres fill as large proportion of the space as possible. In the case where the space is Euclidean this problem is well known, but the case of non-Euclidean metrics is studied much worse. However there are some applied problems, which lead us to use other special non-Euclidean metrics. For instance such statements appear in the logistics when we need to locate a given number of commercial facilities and to maximize the overall service area. Notice, that we consider the optimal packing problem in the case, where P is a multiply-connected domain. The special algorithm based on optical-geometric approach is suggested and implemented. The results of numerical experiment are presented and discussed.

## Keywords

Optimal packing problem Equal circles Non-Euclidean space Multiply-connected domain Numerical algorithm Computational experiment

## References

1. 1.
Conway, J., Sloane, N.: Sphere Packing, Lattices and Groups. Springer Science and Business Media, New York (1999)
2. 2.
Szabo, P., Specht, E.: Packing up to 200 equal circles in a square. In: Torn, A., Zilinskas, J. (eds.) Models and Algorithms for Global Optimization. Optimization and Its Applications, vol. 4, pp. 141–156. Springer, Heidelberg (2007)
3. 3.
Kellerer, H., Pferschy, U., Pisinger, D.: Knapsack Problems. Springer, Berlin (2004)
4. 4.
Levenshtein, V.: On bounds for packing in n-dimensional Euclidean space. Sov. Math. Dokl. 20(2), 417–421 (1979)
5. 5.
Garey, M., Johnson, D.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)
6. 6.
Casado, L., Garcia, I., Szabo, P., Csendes, T.: Packing equal circles in a square ii. New results for up to 100 circles using the tamsass-pecs algorithm. In: Giannessi, F., Pardalos, P., Rapcsac, T. (eds.) Optimization Theory: Recent Developments from Matrahaza, vol. 59, pp. 207–224. Kluwer Academic Publishers, Dordrecht (2001)
7. 7.
Markot, M., Csendes, T.: A new verified optimization technique for the “packing circles in a unit square” problems. SIAM J. Optim. 16(1), 193–219 (2005)
8. 8.
Goldberg, M.: Packing of 14, 16, 17 and 20 circles in a circle. Math. Mag. 44(3), 134–139 (1971)
9. 9.
Graham, R., Lubachevsky, B., Nurmela, K., Ostergard, P.: Dense packings of congruent circles in a circle. Discrete Math. 181(1–3), 139–154 (1998)
10. 10.
Lubachevsky, B., Graham, R.: Curved hexagonal packings of equal disks in a circle. Discrete Comput. Geom. 18, 179–194 (1997)
11. 11.
Birgin, E., Gentil, J.: New and improved results for packing identical unitary radius circles within triangles, rectangles and strips. Comput. Oper. Res. 37(7), 1318–1327 (2010)
12. 12.
Galiev, S., Lisafina, M.: Linear models for the approximate solution of the problem of packing equal circles into a given domain. Eur. J. Oper. Res. 230(3), 505–514 (2013)
13. 13.
Litvinchev, I., Ozuna, E.: Packing circles in a rectangular container. In: Proceedings of the International Congress on Logistics and Supply Chain, pp. 24–25 (2013)Google Scholar
14. 14.
Litvinchev, I., Ozuna, E.: Integer programming formulations for approximate packing circles in a rectangular container. Math. Probl. Eng. (2014)Google Scholar
15. 15.
Lopez, C., Beasley, J.: A heuristic for the circle packing problem with a variety of containers. Eur. J. Oper. Res. 214, 512–525 (2011)
16. 16.
Lopez, C., Beasley, J.: Packing unequal circles using formulation space search. Comput. Oper. Res. 40, 1276–1288 (2013)
17. 17.
Pedroso, J., Cunha, S., Tavares, J.: Recursive circle packing problems. Int. Trans. Oper. Res. 23(1), 355–368 (2014)
18. 18.
Andrade, R., Birgin, E.: Symmetry-breaking constraints for packing identical rectangles within polyhedra. Optim. Lett. 7(2), 375–405 (2013)
19. 19.
Lempert, A., Kazakov, A.: On mathematical models for optimization problem of logistics infrastructure. Int. J. Artif. Intell. 13(1), 200–210 (2015)Google Scholar
20. 20.
Coxeter, H.S.M.: Arrangements of equal spheres in non-Euclidean spaces. Acta Math. Acad. Scientiarum Hung. 5(3), 263–274 (1954)
21. 21.
Boroczky, K.: Packing of spheres in spaces of constant curvature. Acta Math. Acad. Scientiarum Hung. 32(3), 243–261 (1978)
22. 22.
Szirmai, J.: The optimal ball and horoball packings of the coxeter tilings in the hyperbolic 3-space. Beitr. Algebra Geom. 46(2), 545–558 (2005)
23. 23.
Szirmai, J.: The optimal ball and horoball packings to the coxeter honeycombs in the hyperbolic d-space. Beitr. Algebra Geom. 48(1), 35–47 (2007)
24. 24.
Szirmai, J.: A candidate for the densest packing with equal balls in thurston geometries. Beitr. Algebra Geom. 55(2), 441–452 (2014)
25. 25.
Preparata, F., Shamos, M.: Computational Geometry. An Introduction. Springer, New York (1985)
26. 26.
Lempert, A., Kazakov, A., Bukharov, D.: Mathematical model and program system for solving a problem of logistic object placement. Autom. Remote Control 76(8), 1463–1470 (2015)
27. 27.
Kazakov, A., Lempert, A., Bukharov, D.: On segmenting logistical zones for servicing continuously developed consumers. Autom. Remote Control 74(6), 968–977 (2013)
28. 28.
Kazakov, A., Lempert, A.: An approach to optimization in transport logistics. Autom. Remote Control 72(7), 1398–1404 (2011)
29. 29.
Specht, E.: Packomania. http://www.packomania.com/. Accessed 28 Oct 2015
30. 30.
Stoyan, Y., Yaskov, G.: Packing congruent spheres into a multi-connected polyhedral domain. Int. Trans. Oper. Res. 20(1), 79–99 (2013)