Approximating the chromatic polynomial is as hard as computing it exactly
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We show that for any non-real algebraic number q such that |q-1|>1 or (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|≤ 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.
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