Approximating the chromatic polynomial is as hard as computing it exactly

Abstract

We show that for any non-real algebraic number q, such that $$|q-1|>1$$ | q - 1 | > 1 or $$\Re(q)>\frac{3}{2}$$ ℜ ( q ) > 3 2 it is #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 #P-hardness for all non-real algebraic q on the family of all graphs. We, moreover, prove several hardness results for q, such that $$|q-1|\leq 1$$ | q - 1 | ≤ 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 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.