On the Complexity of Matrix Rank and Rigidity

  • Meena Mahajan
  • Jayalal Sarma M.N.
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4649)


We revisit a well studied linear algebraic problem, computing the rank and determinant of matrices, in order to obtain completeness results for small complexity classes. In particular, we prove that computing the rank of a class of diagonally dominant matrices is complete for L. We show that computing the permanent and determinant of tridiagonal matrices over ℤ is in GapNC 1 and is hard for NC 1. We also initiate the study of computing the rigidity of a matrix: the number of entries that needs to be changed in order to bring the rank of a matrix below a given value. It is NP-hard over \(\mathbb{F}_2\) and we prove that some restricted versions characterize small complexity classes. We also look at a variant of rigidity where there is a bound on the amount of change allowed. Using ideas from the linear interval equations literature, we show that this problem is NP-hard over ℚ and that a certain restricted version is NP-complete. Restricting the problem further, we obtain variations which can be computed in PL and are hard for C = L.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allender, E.: Arithmetic circuits and counting complexity classes. In: Krajicek, J. (ed.) Complexity of Computations and Proofs, Quaderni di Matematica, Seconda Universita di Napoli, vol. 13, pp. 33–72 (2004)Google Scholar
  2. 2.
    Allender, E., Ambainis, A., Barrington, D.A.M., Datta, S., LeThanh, H.: Bounded-depth arithmetic circuits: counting and closure. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 149–158. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Allender, E., Beals, R., Ogihara, M.: The complexity of matrix rank and feasible systems of linear equations. Computational Complexity 8(2), 99–126 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bollobas, B.: Modern Graph Theory. GTM, vol. 184. Springer, Heidelberg (1984)Google Scholar
  5. 5.
    Buntrock, G., Damm, C., Hertrampf, U., Meinel, C.: Structure and importance of logspace MOD-classes. Math. Systems Theory 25, 223–237 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Buss, J.F., Frandsen, G.S., Shallit, J.: The computational complexity of some problems of linear algebra. JCSS 58, 572–596 (1999)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Caussinus, H., McKenzie, P., Thérien, D., Vollmer, H.: Nondeterministic NC1 computation. JCSS 57, 200–212 (1998)zbMATHGoogle Scholar
  8. 8.
    Dahl, G.: A note on nonnegative diagonally dominant matrices. Linear Algebra and Applications 317, 217–224 (1999)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Damm, C.: DET=L(#L). Technical Report Informatik-Preprint 8, Fachbereich Informatik der Humboldt–Universität zu Berlin (1991)Google Scholar
  10. 10.
    Deshpande, A.: Sampling-based dimension reduction algorithms. PhD thesis, MIT, May 2007 (expected)Google Scholar
  11. 11.
    Grigoriev, D.Y.: Using the notions of seperability and independence for proving the lower bounds on the circuit complexity. Notes of the Leningrad branch of the Steklov Mathematical Institute, Nauka (1976) (in Russian)Google Scholar
  12. 12.
    Laurent, M.: Matrix completion problems. In: Floudas, C., Pardalos, P. (eds.) The Encyclopedia of Optimization, vol. 3, pp. 221–229. Kluwer, Dordrecht (2001)Google Scholar
  13. 13.
    Lokam, S.V.: Spectral methods for matrix rigidity with applications to size-depth tradeoffs and communication complexity. JCSS 63(3), 449–473 (2001)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Mahajan, M., Sarma, J.M.N.: On the Complexity of Rank and Rigidity. Technical report (2006),
  15. 15.
    Mulmuley, K.: A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica 7, 101–104 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Poljak, S., Rohn, J.: Checking robust nonsingularity is NP-hard. Math. Control Signals Systems 6, 1–9 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Razborov, A.A., Rudich, S.: Natural proofs. JCSS 55(1), 24–35 (1997)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Rohn, J.: Systems of Linear Interval Equations. Linear Algebra and Its Applications 126, 39–78 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Toda, S.: Counting problems computationally equivalent to the determinant. manuscript (1991)Google Scholar
  20. 20.
    Valiant, L.G.: Graph theoretic arguments in low-level complexity. In: Gruska, J. (ed.) Mathematical Foundations of Computer Science 1977. LNCS, vol. 53, pp. 162–176. Springer, Heidelberg (1977)Google Scholar
  21. 21.
    Valiant, L.G.: Why is Boolean complexity theory difficult? In: Proceedings of the London Mathematical Society symposium on Boolean function complexity, pp. 84–94. Cambridge University Press, New York, NY, USA (1992)Google Scholar
  22. 22.
    Vinay, V.: Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In: Selman, A.L. (ed.) Structure in Complexity Theory. LNCS, vol. 223, pp. 270–284. Springer, Heidelberg (1986)Google Scholar
  23. 23.
    Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, Heidelberg (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Meena Mahajan
    • 1
  • Jayalal Sarma M.N.
    • 1
  1. 1.The Institute of Mathematical Sciences, Chennai 600 113India

Personalised recommendations