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 specialisation of the Tutte polynomial Σtijxy . It is shown that, unless NP = RP , many of the relevant coefficients do not even have good randomised approximation schemes. We consider the quasiorder 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 RobertsonSeymour theory, they are in polynomial time, and matrices over finite fields, where they are shown to be NP -hard.