On the complex zeros and the computational complexity of approximating the reliability polynomial

TL;DR

This paper explores the complex zeros of the reliability polynomial, revealing their chaotic behavior and establishing the computational hardness of approximating the polynomial for certain graph classes and values.

Contribution

It establishes a link between the zeros' locations and chaotic dynamics, and proves #P-hardness for evaluating the reliability polynomial at specific points.

Findings

Zeros with modulus > 1 and arbitrary argument exist

Evaluating the reliability polynomial at certain points is #P-hard

Chaotic behavior of rational functions derived from the polynomial

Abstract

In this paper we relate the location of the complex zeros of the reliability polynomial to parameters at which a certain family of rational functions derived from the reliability polynomial exhibits chaotic behaviour. We use this connection to prove new results about the location of reliability zeros. In particular we show that there are zeros with modulus larger than $1$ with essentially any possible argument. We moreover use this connection to show that approximately evaluating the reliability polynomial for planar graphs at a non-positive algebraic number in the unit disk is #P-hard.