Get Permanent Results Faster by Counting Stars

Abstract

We show that $n \times n$ permanents can be computed with $O^*{(2-\varepsilon)^n}$ arithmetic operations over every commutative ring, for $\varepsilon = 10^{-40}$. This solves a problem stated in Knuth's The Art of Computer Programming. Our algorithm implements the classical Binet--Minc identity with dynamic programming and careful bookkeeping. It also gives algebraic branching programs of size $O^*{(2-\varepsilon)^n}$.