# On the hardness of computing the permanent of random matrices

## Abstract

Extending a line of research initiated by Lipton, we study the complexity of computing the permanent of random*n* by*n* matrices with integer values between 0 and*p*−1, for any suitably large prime*p*. 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 13*n*^{3}/*p*, nor can it compute the permanent of at least a\((49n^3 /\sqrt p )\)-fraction of the matrices. As*p* may be expenential in*n*, 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 modulo*p*. 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## Preview

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