Abstract
Given a hypergraph \(\mathcal{H} = (V, \mathcal{F})\) and a [0,1]-valued vector a ∈ [0,1]V, its global rounding is a binary (i.e.,{0,1}-valued) vector α ∈ {0,1}V such that |∑ v ∈ F (a(v) − α(v))|< 1 holds for each \(F \in \mathcal{F}\). We study geometric (or combinatorial) structure of the set of global roundings of a using the notion of compatible set with respect to the discrepancy distance. We conjecture that the set of global roundings forms a simplex if the hypergraph satisfies “shortest-path” axioms, and prove it for some special cases including some geometric range spaces and the shortest path hypergraph of a series-parallel graph.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Asano, T., Katoh, N., Tamaki, H., Tokuyama, T.: The Structure and Number of Global Roundings of a Graph. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 130–138. Springer, Heidelberg (2003)
Asano, T., Katoh, N., Obokata, K., Tokuyama, T.: Matrix Rounding under the Lp-Discrepancy Measure and Its Application to Digital Halftoning. SIAM J. Comput. 32(6), 1423–1435 (2003)
Asano, T., Matsui, T., Tokuyama, T.: Optimal Roundings of Sequences and Matrices. Nordic Journal of Computing 7, 241–256 (2000)
Asano, T., Tokuyama, T.: How to color a checkerboard with a given distribution - matrix rounding achieving low 2×2-discrepancy. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 636–648. Springer, Heidelberg (2001)
Beck, J., Sós, V.T.: Discrepancy Theory. In: Graham, T., Grötshel, M., Lovász, L. (eds.) Handbook of Combinatorics Volume, Elsevier, Amsterdam (1995)
Bohus, G.: On the Discrepancy of 3 Permutations. Random Structuctures & Algorithms 1, 215–220 (1990)
Chazelle, B.: The Discrepancy Method .Cambridge U. Press, Cambridge (2000)
Doerr, B.: Lattice Approximation and Linear Discrepancy of Totally Unimodular Matrices. In: Proc. 12th ACM-SIAM SODA, pp. 119–125 (2001)
Jansson, J., Tokuyama, T.: Semi-Balanced Coloring of Graphs– 2-Colorings Based on a Relaxed Discrepancy Condition (to appear in Graph and Combinatorics)
Juvan, M., Mohar, B., Thomas, R.: List edge-coloring of series-parallel graphs. The Electoronic Journal of Combinatorics 6, 1–6 (1999)
Matoušek, J.: Geometric Discrepancy, Algorithms and Combinatorics, vol. 18. Springer, Heidelberg (1999)
Rödl, V., Winkler, P.: Concerning a Matrix Approximation Problem, Crux Mathmaticorum, pp. 76-79 (1990)
Sadakane, K., Takki-Chebihi, N., Tokuyama, T.: Combinatorics and Algorithms on Low-Discrepancy Roundings of a Real Sequence. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 166–177. Springer, Heidelberg (2001)
Takki-Chebihi, N., Tokuyama, T.: Enumerating Roundings for an Outerplanar Graph. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 425–433. Springer, Heidelberg (2003)
Zhou, X., Matsuo, Y., Nishizeki, T.: List Total Coloring of Series-Parallel Graphs. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 172–181. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Asano, T., Katoh, N., Tamaki, H., Tokuyama, T. (2004). On Geometric Structure of Global Roundings for Graphs and Range Spaces. In: Hagerup, T., Katajainen, J. (eds) Algorithm Theory - SWAT 2004. SWAT 2004. Lecture Notes in Computer Science, vol 3111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27810-8_39
Download citation
DOI: https://doi.org/10.1007/978-3-540-27810-8_39
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22339-9
Online ISBN: 978-3-540-27810-8
eBook Packages: Springer Book Archive