Abstract
We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d:
i) Given n points in ℝd, compute their minimum enclosing cylinder.
ii) Given two n-point sets in ℝd, decide whether they can be separated by two hyperplanes.
iii) Given a system of n linear inequalities with d variables, find a maximum size feasible subsystem.
We show that (the decision versions of) all these problems are W[1]-hard when parameterized by the dimension d. Our reductions also give a n Ω(d)-time lower bound (under the Exponential Time Hypothesis).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Armaldi, E., Kann, V.: The complexity and approximability of finding maximum feasible subsystems of linear relations. Theoretical Computer Science 147, 181–210 (1995)
Aronov, B., Har-Peled, S.: On approximating the depth and related problems. SIAM J. Comput. 38(3), 899–921 (2008)
Arora, S., Babai, L., Stern, J., Sweedyk, Z.: The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. Syst. Sci. 54(2), 317–331 (1997)
Bremner, D., Chen, D., Iacono, J., Langerman, S., Morin, P.: Output-sensitive algorithms for tukey depth and related problems. Statistics and Computing 18(3), 259–266 (2008)
Bădoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proc. 34th Annual ACM Symposium on Theory of Computing, pp. 250–257 (2002)
Cabello, S., Giannopoulos, P., Knauer, C., Marx, D., Rote, G.: Geometric clustering: fixed-parameter tractability and lower bounds with respect to the dimension. ACM Transactions on Algorithms (to appear, 2009)
Cabello, S., Giannopoulos, P., Knauer, C., Rote, G.: Geometric clustering: fixed-parameter tractability and lower bounds with respect to the dimension. In: Proc. 19th Ann. ACM-SIAM Sympos. Discrete Algorithms, pp. 836–843 (2008)
Chan, T.M.: A (slightly) faster algorithm for Klee’s measure problem. In: Proc. 24th Annual Symposium on Computational Geometry, pp. 94–100 (2008)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, Heidelberg (1999)
Giannopoulos, P., Knauer, C., Whitesides, S.: Parameterized complexity of geometric problems. Computer Journal 51(3), 372–384 (2008)
Impagliazzo, R., Paturi, R.: On the complexity of k-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)
Langerman, S., Morin, P.: Covering things with things. Discrete & Computational Geometry 33(4), 717–729 (2005)
Megiddo, N.: On the complexity of polyhedral separability. Discrete & Computational Geometry 3, 325–337 (1988)
Megiddo, N.: On the complexity of some geometric problems in unbounded dimension. J. Symb. Comput. 10, 327–334 (1990)
Varadarajan, K., Venkatesh, S., Ye, Y., Zhang, J.: Approximating the radii of point sets. SIAM J. Comput. 36(6), 1764–1776 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Giannopoulos, P., Knauer, C., Rote, G. (2009). The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension. In: Chen, J., Fomin, F.V. (eds) Parameterized and Exact Computation. IWPEC 2009. Lecture Notes in Computer Science, vol 5917. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11269-0_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-11269-0_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11268-3
Online ISBN: 978-3-642-11269-0
eBook Packages: Computer ScienceComputer Science (R0)