Zero-One Permanent is #P-Complete, A Simpler Proof

Abstract

In 1979, Valiant proved that computing the permanent of a 01-matrix is #PComplete. In this paper we present another proof for the same result. Our proof uses "black box" methodology, which facilitates its presentation. We also prove that deciding whether the permanent is divisible by a small prime is #P-Hard. We conclude by proving that a polynomially bounded function can not be #P-Complete under "reasonable" complexity assumptions. 1 Introduction The permanent has been the object of study by mathematicians since first appearing in the work of Cauchy and Binet in 1812. Despite its syntactical similarity to the determinant, no efficient procedure for computing the permanent is known. In 1979, Valiant provided a reason for this difficulty. In a landmark paper ([Val79a]) he showed that the permanent function is complete for the class #P of enumeration problems. Moreover, Valiant proved that even for 01-matrices, the problem remains #P-Complete. Valiant's proof has two parts. In the fir..