Chromatic, flow and reliability polynomials: The complexity of their coefficients
We study the complexity of computing the coefficients of three classical polynomials, namely the chromatic, flow and reliability polynomials of a graph. Each of these is a specialization of the Tutte polynomial ∑tijxiyj. It is shown that, unless NP = RP, many of the relevant coefficients do not even have good randomized approximation schemes. We consider the quasi-order induced by approximation reducibility and highlight the pivotal position of the coefficient t10 = t01, otherwise known as the beta invariant. Our nonapproximability results are obtained by showing that various decision problems based on the coefficients are NP-hard. A study of such predicates shows a significant difference between the case of graphs, where, by Robertson-Seymour theory, they are computable in polynomial time, and the case of matrices over finite fields, where they are shown to be NP-hard.
Publication Source (Journal or Book title)
Combinatorics Probability and Computing
Oxley, J., & Welsh, D. (2002). Chromatic, flow and reliability polynomials: The complexity of their coefficients. Combinatorics Probability and Computing, 11 (4), 403-426. https://doi.org/10.1017/S0963548302005175