Abstract
In this paper, we first study the problem of rounding a real-valued matrix into an integer-valued matrix to minimize an L p-discrepancy measure between them. To define the L p-discrepancy measure, we introduce a family F of regions (rigid submatrices) of the matrix, and consider a hypergraph defined by the family. The difficulty of the problem depends on the choice of the region family F. We first investigate the rounding problem by using integer programming problems with convex piecewise-linear objective functions. Then, we propose “laminar family” for constructing a practical and well-solvable class of F. Indeed, we show that the problem is solvable in polynomial time if F is the union of two laminar families. We shall present experimental results. We also give some nontrivial upper bounds for the L p-discrepancy. We then foucs on the number of global roundings defined on a hypergraph H G = (V, P G ) which corresponds to a set of shortest paths for a weighted graph G = (V,E). For a given real assignment a on V satisfying 0 ≤ a(v) ≤ 1, a global rounding a with respect to H G is a binary assignment satisfying that |Σν ∈ F a(ν) − α(ν)| < 1 for every F ∈ P G . We conjecture that there are at most |V| + 1 global roundings for H G
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R. Ahuja, T. Magnanti, and J. Orlin. Network Flows, Theory Algorithms and Applications. Prentice Hall, 1993.
T. Asano, T. Matsui, and T. Tokuyama. Optimal roundings of sequences and matrices. Nordic Journal of Computing, 7:241–256, 2000.
Tetsuo Asano, Naoki Katoh, Hisao Tamaki, and Takeshi Tokuyama. The structure and number of global roundings of a graph. Proc. of COCOON 2003, LNCS 2697, pages 130–138, 2003.
Tetsuo Asano, Naoki Katoha, Koji Obokata, and Takeshi Tokuyama. Matrix rounding under the lp-discrepancy measure and its application to digital halftoning. SODA 2002 (also to appear in SIAM J. Cornput.), pages 896–904, 2002.
Z. Baranyai. On the factorization of the complete uniform hypergraphs. Infinite and Finite Sets, (A. Hanaj, R. Rado and V. T. Sós, eds.), Colloq. Math. Soc. J’anos Bolyai, 10:91–108, 1974.
B. E. Bayer. An optimum method for two-level rendition of continuous-tone pictures. Conference Record, IEEE International Conf. on Communications 1:26-11–26-15, 1973.
J. Beck and V. T. Süs. Discrepancy Theory, in Handbook of Combinatorics, volume 11. Elsevier, 1995.
R. W. Floyd and L. Steinberg. An adaptive algorithm for spatial gray scale. SID 75 Digest, Society for Information Display, pages 36–37, 1975.
A. Ghoulia-Houri. Characterisation des matrices totalement unimodulaires. C.R. Acad/ Sci. Paris, 254:1192–1194, 1962.
D.S. Hochbaum and J.G. Shanthikumar. Nonlinear separable optimization is not much harder than linear optimization. Journal of ACM, 37:843–862, 1990.
http://www.algorithmic-solutions.com/as_html/products/products.html. LEDA homepage. Algorithmic Solutions Software GmbH, 2003.
D.E. Knuth. D.e. knuth. ACM Trans. Graphics, 6:245–273, 1987.
J. O. Limb. Design of dither waveforms for quantized visual signals. Bell Syst. Tech. J., 48–7:2555–2582, 1969.
Q. Lin. Halftone image quality analysis based on a human vision model. Proceedings of SPIE, 1913:378–389, 1993.
B. Lippel and M. Kurland. The effect of dither on luminance quantization of pictures. IEEE Trans. Commun. Tech., COM-19:879–888, 1971.
M. Minoux. Solving integer minimum cost flows with separable cost objective polynomially. Mathematical Programming Study, 46:237–239, 1986.
K. Sadakane, N. Takki-Chebihi, and T. Tokuyama. Combinatorics and algorithms on low-discrepancy roundings of a real sequence. Proc. 28th ICALP, Springer LNCS 2076, pages 166–177, 2001.
A. Schrijver. Theory of Linear and Integer Programming. Wiley-Interscience Series in Discrete Mathematics. John Wiley and Sons, 1986.
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Katoh, N. (2005). Matrix Rounding and Related Problems with Application to Digital Halftoning. In: Cagnol, J., Zolésio, JP. (eds) System Modeling and Optimization. CSMO 2003. IFIP International Federation for Information Processing, vol 166. Springer, Boston, MA. https://doi.org/10.1007/0-387-23467-5_4
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DOI: https://doi.org/10.1007/0-387-23467-5_4
Publisher Name: Springer, Boston, MA
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