Abstract
We consider the problem of packing a set of circles into a minimum number of unit square bins. We give an asymptotic approximation scheme (APTAS) when we have resource augmentation in one dimension, that is, we may use bins of height 1 + γ, for some small γ > 0. As a corollary, we also obtain an APTAS for the circle strip packing problem, whose objective is to pack a set of circles into a strip of unit width and minimum height. These are the first approximation schemes for these problems. Our algorithm is based on novel ideas of iteratively separating small and large items, and may be extended to more general packing problems. For example, we also obtain APTAS’s for the corresponding problems of packing d-dimensional spheres under the L p -norm.
This work was partially supported by CNPq (grants 303987/2010-3, 306860/2010-4, 477203/2012-4, and 477692/2012-5), FAPESP (grants 2010/20710-4, 2013/02434-8, 2013/03447-6, and 2013/21744-8), and Project MaClinC of NUMEC at USP, Brazil.
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References
Bansal, N., Caprara, A., Sviridenko, M.: A New Approximation Method for Set Covering Problems, with Applications to Multidimensional Bin Packing. SIAM J. on Computing 39(4), 1256–1278 (2010)
Bansal, N., Correa, J.R., Kenyon, C., Sviridenko, M.: Bin Packing in Multiple Dimensions: Inapproximability Results and Approximation Schemes. Mathematics of Operations Research 31(1), 31–49 (2006)
Bansal, N., Han, X., Iwama, K., Sviridenko, M., Zhang, G.: A harmonic algorithm for the 3D strip packing problem. SIAM J. on Computing 42(2), 579–592 (2013)
Bansal, N., Khan, A.: Improved Approximation Algorithm for Two-Dimensional Bin Packing. In: SODA 2014, pp. 13–25 (2014)
Basu, S., Pollack, R., Roy, M.F.: On the Combinatorial and Algebraic Complexity of Quantifier Elimination. J. ACM 43(6), 1002–1045 (1996)
Birgin, E.G., Gentil, J.M.: New and improved results for packing identical unitary radius circles within triangles, rectangles and strips. Computers & Op. Research 37(7), 1318–1327 (2010)
Caprara, A.: Packing d-dimensional bins in d stages. Mathematics of Operations Research 33(1), 203–215 (2008)
Chung, F., Garey, M., Johnson, D.: On Packing Two-Dimensional Bins. SIAM Journal on Algebraic Discrete Methods 3(1), 66–76 (1982)
Coffman, J.E.G., Csirik, J., Galambos, G., Martello, S., Vigo, D.: Bin Packing Approximation Algorithms: Survey and Classification. In: Pardalos, P.M., Du, D.Z., Graham, R.L. (eds.) Handbook of Combinatorial Optimization, pp. 455–531. Springer, New York (2013)
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
Demaine, E.D., Fekete, S.P., Lang, R.J.: Circle Packing for Origami Design Is Hard. In: Proc. of the 5th Inter. Conference on Origami in Science, pp. 609–626 (2010)
Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within 1 + ε in linear time. Combinatorica 1(4), 349–355 (1981)
George, J.A., George, J.M., Lamar, B.W.: Packing different-sized circles into a rectangular container. European J. of Operational Research 84(3), 693–712 (1995)
Grigor’ev, D.Y., Vorobjov Jr., N.N.: Solving systems of polynomial inequalities in subexponential time. Journal of Symbolic Computation 5(1-2), 37–64 (1988)
Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: Models and methodologies. Advances in Operations Research 2009, 1–22 (2009)
Kenyon, C., Rémila, E.: A Near-Optimal Solution to a Two-Dimensional Cutting Stock Problem. Mathematics of Operations Research 25(4), 645–656 (2000)
Kohayakawa, Y., Miyazawa, F., Raghavan, P., Wakabayashi, Y.: Multidimensional Cube Packing. Algorithmica 40(3), 173–187 (2004)
Meir, A., Moser, L.: On packing of squares and cubes. Journal of Combinatorial Theory 5(2), 126–134 (1968)
Miyazawa, F., Wakabayashi, Y.: Approximation algorithms for the orthogonal z-oriented three-dimensional packing problem. J. on Comp. 29(3), 1008–1029 (2000)
Szabó, P.G., Markót, M.C., Csendes, T., Specht, E., Casado, L., García, I.: New-Approaches-to-Circle-Packing-in-a-Square-Book. Springer (2007)
Tarski, A.: A decision method for elementary algebra and geometry. University of California Press (1951)
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Miyazawa, F.K., Pedrosa, L.L.C., Schouery, R.C.S., Sviridenko, M., Wakabayashi, Y. (2014). Polynomial-Time Approximation Schemes for Circle Packing Problems. In: Schulz, A.S., Wagner, D. (eds) Algorithms - ESA 2014. ESA 2014. Lecture Notes in Computer Science, vol 8737. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_59
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DOI: https://doi.org/10.1007/978-3-662-44777-2_59
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