Approximating the chromatic polynomial is as hard as computing it exactly

TL;DR

This paper proves that approximating the chromatic polynomial at certain complex numbers is computationally as hard as computing it exactly, establishing P-hardness for a broad class of cases on planar and general graphs.

Contribution

It demonstrates P-hardness of approximation for the chromatic polynomial at specific non-real algebraic points, extending hardness results to the partition function of the random cluster model.

Findings

P-hardness for approximating the chromatic polynomial at certain complex points.

Hardness results extend to the partition function of the random cluster model.

Approximate computation at these points implies exact computation, which is P-hard.

Abstract

We show that for any non-real algebraic number $q$ such that $|q-1|>1$ or $\Re(q)>\frac{3}{2}$ it is \textsc{\#P}-hard to compute a multiplicative (resp. additive) approximation to the absolute value (resp. argument) of the chromatic polynomial evaluated at $q$ on planar graphs. This implies \textsc{\#P}-hardness for all non-real algebraic $q$ on the family of all graphs. We moreover prove several hardness results for $q$ such that $|q-1|\leq 1$. Our hardness results are obtained by showing that a polynomial time algorithm for approximately computing the chromatic polynomial of a planar graph at non-real algebraic $q$ (satisfying some properties) leads to a polynomial time algorithm for \emph{exactly} computing it, which is known to be hard by a result of Vertigan. Many of our results extend in fact to the more general partition function of the random cluster model, a well known reparametrization of the Tutte polynomial.