Computing permanents of complex diagonally dominant matrices and tensors

TL;DR

This paper presents a quasi-polynomial time algorithm for approximating the permanent of certain complex diagonally dominant matrices and tensors, with applications to counting perfect matchings in hypergraphs.

Contribution

It introduces a novel approximation method for permanents of complex diagonally dominant matrices and tensors, extending previous results to multidimensional cases.

Findings

Approximation of permanents within any relative error in quasi-polynomial time.

Extension of results to multidimensional tensor permanents.

Application to weighted counting of perfect matchings in hypergraphs.

Abstract

We prove that for any $\lambda > 1$, fixed in advance, the permanent of an $n \times n$ complex matrix, where the absolute value of each diagonal entry is at least $\lambda$ times bigger than the sum of the absolute values of all other entries in the same row, can be approximated within any relative error $0 < \epsilon < 1$ in quasi-polynomial $n^{O(\ln n - \ln \epsilon)}$ time. We extend this result to multidimensional permanents of tensors and discuss its application to weighted counting of perfect matchings in hypergraphs.