Computing the permanent of (some) complex matrices

TL;DR

This paper introduces a deterministic algorithm for approximating the permanent of certain complex matrices with high accuracy in quasi-polynomial time, extending to hafnians and multidimensional permanents.

Contribution

It provides a novel deterministic approximation method for the permanent of matrices close to all-ones matrices, with potential extensions to related combinatorial quantities.

Findings

Computes permanent within relative error epsilon efficiently for matrices close to all-ones.

Algorithm runs in n^{O(ln n - ln epsilon)} time, which is quasi-polynomial.

Extensible to hafnians and multidimensional permanents.

Abstract

We present a deterministic algorithm, which, for any given 0< epsilon < 1 and an nxn real or complex matrix A=(a_{ij}) such that | a_{ij}-1| < 0.19 for all i, j computes the permanent of A within relative error epsilon in n^{O(ln n -ln epsilon)} time. The method can be extended to computing hafnians and multidimensional permanents.