Computational Optimization and Applications

, Volume 51, Issue 3, pp 1397–1408 | Cite as

Two complexity results on c-optimality in experimental design

Article

Abstract

Finding a c-optimal design of a regression model is a basic optimization problem in statistics. We study the computational complexity of the problem in the case of a finite experimental domain. We formulate a decision version of the problem and prove its \(\boldsymbol{\mathit{NP}}\)-completeness. We provide examples of computationally complex instances of the design problem, motivated by cryptography. The problem, being \(\boldsymbol{\mathit{NP}}\)-complete, is then relaxed; we prove that a decision version of the relaxation, called approximate c-optimality, is P-complete. We derive an equivalence theorem for linear programming: we show that the relaxed c-optimality is equivalent (in the sense of many-one LOGSPACE-reducibility) to general linear programming.

Keywords

Linear programming Optimal design NP-completeness P-completeness 

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References

  1. 1.
    Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. Math. 160, 781–793 (2004) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Atkinson, A., Donev, A., Tobias, R.: Optimum Experimental Designs with SAS. Oxford University Press, Oxford (2007) MATHGoogle Scholar
  3. 3.
    Dobkin, D., Lipton, R., Reiss, S.: Linear programming is log-space hard for P. Inf. Process. Lett. 8(2), 96–97 (1979) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Garey, M., Johnson, D.: Computers and Intractability. A Guide to the Theory of NP-Completeness. Freeman, New York (1979) MATHGoogle Scholar
  5. 5.
    Greenlaw, R., Hoover, H., Ruzzo, W.: Limits to Parallel Computation. P-completeness Theory. Oxford University Press, Oxford (1995) MATHGoogle Scholar
  6. 6.
    Harman, R., Jurík, T.: Computing c-optimal experimental designs using the simplex method of linear programming. Comput. Stat. Data Anal. 53, 247–254 (2008) MATHCrossRefGoogle Scholar
  7. 7.
    Khachyian, L.: A polynomial algorithm for linear programming. Dokl. Soviet Acad. Sci. 244(5), 1093–1096 (1979) Google Scholar
  8. 8.
    Papadimitriou, C.: Computational Complexity. Addison-Wesley, Longman (1995) Google Scholar
  9. 9.
    Pázman, A.: Foundations of Optimum Experimental Design. Reidel, Dordrecht (1986) MATHGoogle Scholar
  10. 10.
    Pukelsheim, F., Rieder, S.: Efficient rounding in approximate designs. Biometrika 79, 763–770 (1992) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of EconometricsUniversity of Economics, PraguePragueCzech Republic
  2. 2.Faculty of Mathematics and Physics, Department of Applied MathematicsCharles University, PraguePragueCzech Republic

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