Complexity and Approximability of Hyperplane Covering Problems

Michael Khachay

Abstract

The well known N.Megiddo complexity result for Point Cover Problem on the plane is extended onto $d$-dimensional space (for any fixed $d$). It is proved that Min-$d$PC problem is $L$-reducible to Min-$(d+1)$PC problem, therefore for any fixed $d>1$ there is no PTAS for Min-$d$PC problem, unless $P=NP.$

References

  1. Agarwal P.K. and Procopiuc C.M. Exact and approximation algorithms for clustering, Algorithmica, No. 33, 201-206. (2002).
  2. Langerman S. and P. Morin. Covering things with things, Discrete Computat. Geom., 717- 729, (2005).
  3. Khachai M. Computational complexity of recognition learning procedures in the class of piecewise-linear committee decision rules, Automation and Remote Control, 71, No. 3, 528- 539, (2010).
  4. Vazirany V. Approximation algorithms, Springer (2001).
  5. Johnson D. Approximation algorithms for combinatorial problems, Journal of Computer and System Sciences, 9, No. 3, 256-278, (1974).
  6. Lovász L. On the ratio of integer and fractional covers, Discrete Mathematics. No. 13, 383- 390, (1975).
  7. Feige U. A Threshold of ln n for Approximating Set Cover, Journal of the ACM, 45, No. 4, 634-652, (1998).
  8. Megiddo N. and Tamir A. On the complexity of locating linear facilities in the plane, Operations research letters. 1, No. 5, 194-197, (1982).
  9. Papadimitriou C. and Yannakakis M. Optimization, approximation, and complexity classes, J. Comput. System Sci., 43, No. 3, 425-440, (1991).
  10. Papadimitriou C. Computational Complexity, Addison-Wesley, (1995).
  11. Khachai M. and Poberii M.Computational complexity of combinatorial problems related to piecewise linear committee pattern recognition learning procedures, Pattern Recognition and Image Analysis. 22, No. 2, 278-290, (2012).
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Paper Citation


in Harvard Style

Khachay M. (2013). Complexity and Approximability of Hyperplane Covering Problems . In Proceedings of the 4th International Workshop on Image Mining. Theory and Applications - Volume 1: IMTA-4, (VISIGRAPP 2013) ISBN 978-989-8565-50-1, pages 109-113. DOI: 10.5220/0004394601090113


in Bibtex Style

@conference{imta-413,
author={Michael Khachay},
title={Complexity and Approximability of Hyperplane Covering Problems},
booktitle={Proceedings of the 4th International Workshop on Image Mining. Theory and Applications - Volume 1: IMTA-4, (VISIGRAPP 2013)},
year={2013},
pages={109-113},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0004394601090113},
isbn={978-989-8565-50-1},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 4th International Workshop on Image Mining. Theory and Applications - Volume 1: IMTA-4, (VISIGRAPP 2013)
TI - Complexity and Approximability of Hyperplane Covering Problems
SN - 978-989-8565-50-1
AU - Khachay M.
PY - 2013
SP - 109
EP - 113
DO - 10.5220/0004394601090113