computational complexity

, Volume 6, Issue 2, pp 101–132 | Cite as

On the hardness of computing the permanent of random matrices

  • Uriel Feige
  • Carsten Lund


Extending a line of research initiated by Lipton, we study the complexity of computing the permanent of randomn byn matrices with integer values between 0 andp−1, for any suitably large primep. Previous to our work, it was shown hard to compute the permanent of half these matrices (by Gemmell and Sudan), and to enumerate for any matrix a polynomial number of options for its permanent (by Cai and Hemachandra, and by Toda). We show that unless the polynomial-time hierarchy collapses to its second level, no polynomial time algorithm can compute the permanent of every matrix with probability at least 13n3/p, nor can it compute the permanent of at least a\((49n^3 /\sqrt p )\)-fraction of the matrices. Asp may be expenential inn, these represent very low success probabilities for any efficient algorithm that attempts to compute the permanent. For 0/1 matrices, our results show that their permanents cannot be guessed with probability greater than\(1/2^{n^{1 - \in } }\).

We also show that it is hard to get even partial information about the value of the permanent modulop. For random matrices we show that any balanced polynomial-time 0/1 predicate (e.g., the least significant bit, the parity of all the bits, the quadratic residuosity character) cannot be guessed with probability significantly greater than 1/2 (unless the polynomial-time hierarchy collapses). This result extends to showing simultaneous hardness for linear size groups of bits.

Key words

Permanent computational complexity heuristics interactive proof systems 

Subject classifications

15A15 68Q15 68Q25 


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Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  • Uriel Feige
    • 1
  • Carsten Lund
    • 2
  1. 1.Department of Applied MathematicsThe Weizmann InstituteRehovotIsrael
  2. 2.AT&T ResearchMurray Hill

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