Skip to main content
Springer Nature Link
Log in

Exponential Inapproximability of Selecting a Maximum Volume Sub-matrix

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Given a matrix A∈ℝm×n (n vectors in m dimensions), and a positive integer k<n, we consider the problem of selecting k column vectors from A such that the volume of the parallelepiped they define is maximum over all possible choices. We prove that there exists δ<1 and c>0 such that this problem is not approximable within 2−ck for k=δn, unless P=NP.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM 45(1), 70–122 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boutsidis, C., Drineas, P., Magdon-Ismail, M.: Near-optimal column based matrix reconstruction. (2011). arXiv:1103.0995v1

  4. Boutsidis, C., Mahoney, M.W., Drineas, P.: An improved approximation algorithm for the column subset selection problem. In: SODA’09: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 968–977 (2009)

    Google Scholar 

  5. Chan, T.F.: Rank revealing QR factorizations. Linear Algebra Appl. 88/89, 67–82 (1987)

    Article  Google Scholar 

  6. Chan, T.F., Hansen, P.: Low-rank revealing QR factorizations. Numer. Linear Algebra Appl. 1, 33–44 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chandrasekaran, S., Ipsen, I.C.F.: On rank-revealing factorizations. SIAM J. Matrix Anal. Appl. 15, 592–622 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Çivril, A., Magdon-Ismail, M.: On selecting a maximum volume sub-matrix of a matrix and related problems. Theor. Comput. Sci. 410(47–49), 4801–4811 (2009)

    Article  MATH  Google Scholar 

  9. de Hoog, F.R., Mattheijb, R.M.M.: Subset selection for matrices. Linear Algebra Appl. 422, 349–359 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deshpande, A., Rademacher, L.: Efficient volume sampling for row/column subset selection. In: FOCS’10: Proceedings of 51st Annual Symposium on Foundations of Computer Science, pp. 329–338 (2010)

    Chapter  Google Scholar 

  11. Deshpande, A., Rademacher, L., Vempala, S., Wang, G.: Matrix approximation and projective clustering via volume sampling. Theory Comput. 2(1), 225–247 (2006)

    Article  MathSciNet  Google Scholar 

  12. Deshpande, A., Vempala, S.: Adaptive sampling and fast low-rank matrix approximation. In: RANDOM’06: 10th International Workshop on Randomization and Computation, pp. 292–303 (2006)

    Google Scholar 

  13. Feige, U.: A threshold of lnn for approximating set cover. J. ACM 45(4), 634–652 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Golub, G.H., Klema, V., Stewart, G.W.: Rank degeneracy and least squares problems. Technical report, Dept. of Computer Science, Univ. of Maryland, 1976

  15. Golub, G.H., Loan, C.V.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  16. Goreinov, S.A., Tyrtyshnikov, E.E.: The maximal-volume concept in approximation by low-rank matrices. In: Contemporary Mathematics, vol. 280, pp. 47–51. AMS, Providence (2001)

    Google Scholar 

  17. Goreinov, S.A., Zamarashkin, N.L., Tyrtyshnikov, E.E.: Pseudo-skeleton approximations by matrices of maximal volume. Mat. Zametki 62, 619–623 (1997)

    Article  MathSciNet  Google Scholar 

  18. Gritzmann, P., Klee, V., Larman, D.G.: Largest j-simplices n-polytopes. Discrete Comput. Geom. 13, 477–515 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gu, M., Eisenstat, S.C.: Efficient algorithms for computing a strong rank-revealing QR factorization. SIAM J. Sci. Comput. 17(4), 848–869 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Comput. Complex. 15, 20–39 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hong, Y.P., Pan, C.T.: Rank-revealing QR factorizations and the singular value decomposition. Math. Comput. 58, 213–232 (1992)

    MathSciNet  MATH  Google Scholar 

  22. Koutis, I.: Parameterized complexity and improved inapproximability for computing the largest j-simplex in a V-polytope. Inf. Process. Lett. 100, 8–13 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Packer, A.: Polynomial-time approximation of largest simplices in V-polytopes. Discrete Appl. Math. 134, 213–237 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pan, C.T.: On the existence and computation of rank-revealing LU factorizations. Linear Algebra Appl. 316(1–3), 199–222 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pan, C.T., Tang, P.T.P.: Bounds on singular values revealed by QR factorizations. BIT Numer. Math. 39, 740–756 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Raz, R.: A parallel repetition theorem. SIAM J. Comput. 27(3), 763–803 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Engineering Department, Melikşah University, Talas, Kayseri, 38280, Turkey

    Ali Çivril

  2. Computer Science Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY, 12180, USA

    Malik Magdon-Ismail

Authors
  1. Ali Çivril
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Malik Magdon-Ismail
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Ali Çivril.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Çivril, A., Magdon-Ismail, M. Exponential Inapproximability of Selecting a Maximum Volume Sub-matrix. Algorithmica 65, 159–176 (2013). https://doi.org/10.1007/s00453-011-9582-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-011-9582-6

Keywords