Subexponential Approximation of the Permanent in Deterministic Polynomial Time

Abstract: We give the first deterministic polynomial time algorithm that approximates the permanent of arbitrary nonnegative rational matrices within a subexponential factor. For a matrix of order $n$, the approximation factor is [ \exp!\left(O!\left(\frac{n(\log\log n)^2}{\log n}\right)\right)=\exp(o(n)). ] All previously known deterministic polynomial time guarantees for unrestricted inputs had approximation factors $\exp(Ω(n))$. Our proof uses convex optimization to tighten an upper bound on the permanent. The bound is based on weighted sums over all matchings in a bipartite graph representing the matrix, and correlations between unmatched vertices control its error. We approximate these sums deterministically using correlation decay and a bound on the effect of vertex deletion.