Chromatic, flow and reliability polynomials: The complexity of their coefficients

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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.

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Combinatorics Probability and Computing

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