On the computational complexity of the permanent

We consider the problem of computing the permanent of an mxn matrix, m≪n, over an arbitrary commutative ring. The permanent is a central problem in the computational complexity of enumeration problems and arises in several applications. We introduce the class of multilinear programs, where the monomials of the factors of any multiplication come from different sets of rows. All the known algorithms to compute the permanent satisfy this property. These programs can be represented by arithmetic circuits with unbounded fan in. We show the following: 1) Given any positive integer k, no constant depth multilinear program with ≪nk arithmetic operations can compute the permanent of an mxn matrix, where m = Ω(γ(n)logn) for any increasing function γ(n). 2) There exists a polynomial-size multilinear program of depth 4 which computes the permanent of an mxn matrix for m = 0(logn/loglogn). 3) Any arbitrary arithmetic circuit that computes the permanent of an mxn matrix with depth ≪3 must be of exponential size for any m nonconstant. Our proofs use, in a nontrivial way, probabilistic techniques to establish several combinatorial facts.