Bounds on the permanent and some applications

TL;DR

This paper establishes new bounds on the permanent of doubly stochastic matrices, improving approximation factors, and applies these bounds to prove a conjecture in the monomer-dimer problem.

Contribution

It introduces improved bounds on the permanent and applies them to prove Friedland's Asymptotic Lower Matching Conjecture.

Findings

Improved deterministic approximation factor for the permanent.

Proved Friedland's Asymptotic Lower Matching Conjecture.

Established new bounds on the permanent of doubly stochastic matrices.

Abstract

We give new lower and upper bounds on the permanent of a doubly stochastic matrix. Combined with previous work, this improves on the deterministic approximation factor for the permanent. We also give a combinatorial application of the lower bound, proving S. Friedland's "Asymptotic Lower Matching Conjecture" for the monomer-dimer problem.