Abstract
In this paper, we study the following disc covering problem: Given a set of discs of various radii on the plane and centers on the grid points, 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 (Asano et al., 2004). We show that if their radii are between two positive constants, then there exists a polynomial time approximation scheme. Our techniques are based on the width-bounded geometric separator recently developed in Fu and Wang (2004), Fu (2006).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alon N, Seymour P, Thomas R (1990) Planar Separator, SIAM J Discr Math 7(2):184–193
Asano T, Brass P, Sasahara S (2004) Disc covering problem with application to digital halftoning, International Conference on Computational Science and Applications, Assisi, Italy, May 14–17, 2004, Proceedings, Part III. (Lecture Notes in Computer Science 3045, Springer 2004, pp. 11–21
Asano T, Matsui T, Tokuyama T (2000) Optimal roundings of sequences and matrices, Nordic Journal of Computing, 3(7): 241–256
Djidjev HN (1982) On the problem of partitioning planar graphs, SIAM Journal on Discrete Mathematics 3(2):229–240
Fu B (2006) Theory and application of width bounded geometric separator. The draft was posted at Electronic Colloquium on Computational Complexity 2005, TR05-13. In: Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science (STACS 2006), February 23–25, 2006, Marseille, France, pp. 277–288
Fu B, Wang W (2004) A \(2^O(n^1-1/d\log n)\)-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
Gazit H (1986) An improved algorithm for separating a planar graph, manuscript, USC
Graham R, Grötschel M, Lovász L (1996) Handbook of combinatorics (volume I), MIT Press
Jadhar S, Mukhopadhyay A (1993) Computing a center of a finite planar set of points in linear time, The 9th annual ACM symposium on computational geometry, pp. 83–90
Lipton RJ, Tarjan R (1979) A separator theorem for planar graph. SIAM J Appl Math 36(1979):177–189
Miller GL, Teng S-H, Vavasis SA (1991) An unified geometric approach to graph separators. In 32nd Annual Symposium on Foundation of Computer Science, IEEE 1991, pp. 538–547
Ostromoukhov V (1993) 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
Ostromoukhov V, Hersch RD (1999) Stochastic clustered-dot dithering. Journal of electronic imaging 4(8):439–445.
Pach J, Agarwal PK (1995) Combinatorial geometry, Wiley-Interscience Publication
Sasahara S, Asano T (2003) Adaptive cluster arrangement for cluster-dot halftoning using bubble packing method, In: Proceeding of 7th Japan joint workshop on algorithms and computation, Sendai, pp. 87–93
Smith WD, Wormald NC (1998) Application of geometric separator theorems, FOCS, pp. 232–243
Spielman DA, Teng SH (1996) Disk packings and planar separators. In: 12th Annual ACM Symposium on Computational Geometry, pp. 349–358
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by Louisiana Board of Regents fund under contract number LEQSF(2004-07)-RD-A-35.
Rights and permissions
About this article
Cite this article
Chen, Z., Fu, B., Tang, Y. et al. A PTAS for a disc covering problem using width-bounded separators. J Comb Optim 11, 203–217 (2006). https://doi.org/10.1007/s10878-006-7132-y
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10878-006-7132-y