Approximating the chromatic polynomial is as hard as computing it exactly

• Published in: Computational complexity Vol. 33; no. 1; p. 1
• Main Authors: Bencs, Ferenc, Huijben, Jeroen, Regts, Guus
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2024; Springer Nature B.V
• Subjects: Algebra; Algorithm Analysis and Problem Complexity; Algorithms; Approximation; Computation; Computational Mathematics and Numerical Analysis; Computer Science; Graphs; Hardness; Mathematical analysis; Partitions (mathematics); Polynomials; P-hardness; Tutte polynomial; 68Q17, primary; approximate counting; planar graphs; 05C31, secondary; chromatic polynomial
• ISSN: 1016-3328; 1420-8954; 1420-8954
• DOI: 10.1007/s00037-023-00247-8

We show that for any non-real algebraic number q , such that | q - 1 | > 1 or 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 | ≤ 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.