Tensor rank is NP-complete

  • Johan Håstad
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 372)


We prove that computing the rank of a three-dimensional tensor over any finite field is NP-complete. Over the rational numbers the problem is NP-hard.

Key words

NP-completeness tensor rank bilinear complexity multiplicative complexity 

AMS classification

68C25 68E99 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AS]
    Alder A. and Strassen V. “On the Algorithmic Complexity of Associative Algebras”, Theoretical Computer Science 15 (1981), pp 201–211.CrossRefGoogle Scholar
  2. [B]
    Bshouty N.H. “A Lower Bound for Matrix Multiplication”, Proceedings 29th Annual IEEE Symposium on Foundations of Computer Science, 1988, pp 64–67.Google Scholar
  3. [C]
    Cook S.A. “On the Complexity of Theorem Proving Procedures”, Proceedings 3rd Annual ACM Symposium on the Theory of Computing, pp 151–159.Google Scholar
  4. [CW]
    Coppersmith D. and Winograd S., “Matrix Multiplication via Arithmetic Progressions”, Proceedings 19th Annual ACM Symposium on the Theory of Computing, pp 1–6.Google Scholar
  5. [GaJo]
    Garey M. R. and Johnson D.S., “Computers and Intractability”, W.H. Freeman and Company, 1979.Google Scholar
  6. [GoJa]
    Gonzalez T. and Ja'Ja' J. “On the Complexity of Computing Bilinear Forms with {0, 1} Constants”, Journal of Computer and Systems Sciences 20, (1980) pp 77–95.CrossRefGoogle Scholar
  7. [S1]
    Strassen V. “The asymptotic spectrum of tensors”, manuscript, 1986.Google Scholar
  8. [S2]
    Strassen V. “Rank and Optimal Computation of Generic Tensors”, Linear Algebra and its applications 52/53 (1983), pp 645–685.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Johan Håstad
    • 1
  1. 1.Royal Institute of TechnologySweden

Personalised recommendations