Abstract
In this paper, we study the following disc covering problem: Given a set of discs of various radii on the plane, find a subset of discs to maximize the area covered by exactly one disc. This problem originates from the application in digital halftoning, with the best known approximation factor being 5.83 [2]. We show that if the maximum radius is no more than a constant times the minimum radius, then there exists a polynomial time approximation scheme. Our techniques are based on the width-bounded geometric separator recently developed in [5,6].
This research is supported by Louisiana Board of Regents fund under contract number LEQSF(2004-07)-RD-A-35.
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References
Alon, N., Seymour, P., Thomas, R.: Planar Separator. SIAM J. Discr. Math. 7(2), 184–193 (1990)
Asano, T., Brass, P., Sasahara, S.: Disc covering problem with application to digital halftoning. In: Laganá, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds.) ICCSA 2004. LNCS, vol. 3045, pp. 11–21. Springer, Heidelberg (2004)
Asano, T., Matsui, T., Tokuyama, T.: Optimal roundings of sequences and matrices. Nordic Journal of Computing 3(7), 241–256 (2000)
Djidjev, H.N.: On the problem of partitioning planar graphs. SIAM Journal on Discrete Mathematics 3(2), 229–240 (1982)
Fu, B.: Theory and Application ofWidth Bounded Geometric Separator, Electronic Colloquium on Computational Complexity, TR05-13 (2005)
Fu, B., Wang, W.: A 2O(n1−1/d log n)-time algorithm for d-dimensional protein folding in the HP-model. In: proceedings of 31st International Colloquium on Automata, Languages and Programming, July 12-26, pp. 630–644 (2004)
Gazit, H.: An improved algorithm for separating a planar graph, manuscript, USC (1986)
Graham, R., Grötschel, M., Lovász, L.: Handbook of combinatorics, vol. I. MIT Press, Cambridge (1996)
Lipton, R.J., Tarjan, R.: A separator theorem for planar graph. SIAM J. Appl. Math. 36, 177–189 (1979)
Miller, G.L., Teng, S.-H., Vavasis, S.A.: An unified geometric approach to graph separators. In: 32nd Annual Symposium on Foundation of Computer Science, pp. 538–547. IEEE, Los Alamitos (1991)
Ostromoukhov, V.: Pseudo-random halftone screening for color and black & white printing. In: Proceedings of the 9th congress on advances in non-impact printing technologies, Yohohama, pp. 579–581 (1993)
Ostromoukhov, V., Hersch, R.D.: Stochastic clustered-dot dithering. Journal of electronic imaging 4(8), 439–445 (1999)
Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley Interscience Publication Publication, Hoboken (1995)
Sasahara, S., Asano, T.: Adaptive cluster arrangement for cluster-dot halftoning using bubble packing method. In: Proceeding of 7th Japan joint workshop on algorithms and computation, Sendai, July 2003, pp. 87–93 (2003)
Smith, W.D., Wormald, N.C.: Application of geometric separator theorems. In: FOCS 1998, pp. 232–243 (1998)
Spielman, D.A., Teng, S.H.: Disk packings and planar separators. In: 12th Annual ACM Symposium on Computational Geometry, pp. 349–358 (1996)
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Chen, Z., Fu, B., Tang, Y., Zhu, B. (2005). A PTAS for a Disc Covering Problem Using Width-Bounded Separators. In: Wang, L. (eds) Computing and Combinatorics. COCOON 2005. Lecture Notes in Computer Science, vol 3595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11533719_50
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DOI: https://doi.org/10.1007/11533719_50
Publisher Name: Springer, Berlin, Heidelberg
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