Favorite Theorems: The Permanent

June Edition

The permanent of a matrix has a simple form

Perm(A)=Σ_σ a_1&sigma(1)a_2&sigma(2)···a_n&sigma(n)

where A is an n×n matrix, a_ij is the element in the ith row and jth column and the sum is taken over all permutations σ of {1,…,n}.

Very similar to the determinant except without the negations and a 0-1 permanent counts the number of perfect matchings on a bipartite graph. But while we can compute the determinant quickly and easily check if a graph has a perfect matching the permanent turns out to have an apparently much higher complexity.

Leslie Valiant, The Complexity of Computing the Permanent, TCS 1979.

Valiant shows that computing the permanent, even for 0-1 matrices, is #P-complete, i.e., as hard as counting the number of solutions for NP problems.

While an important hardness result in its own right, Valiant's theorem becomes even more important with later work. Toda's theorem implies the permanent is hard for the entire polynomial-time hierarchy. The permanent also has two very important properties:

1. The permanent on n×n matrices is easily reducible to solving the permanent on (n-1)×(n-1) matrices.
2. The permanent is a multilinear polynomial on the entries of A.

Those two facts, combined with Valiant's and Toda's theorems, helped the permanent play a critical role in the development of interactive proofs and in some derandomization results, most notably Impagliazzo-Wigderson and Impagliazzo-Kabanets.