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Chromatic, Flow and Reliability Polynomials: The Complexity of their Coefficients

Published online by Cambridge University Press:  06 September 2002

JAMES OXLEY  and
DOMINIC WELSH
JAMES OXLEY
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, 70803, USA (email: oxley@math.LSU.edu)
DOMINIC WELSH
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK (email: dwelsh@maths.ox.ac.uk)
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Abstract

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 Σtijxiyj. 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.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 11 , Issue 4 , July 2002 , pp. 403 - 426
Copyright
2002 Cambridge University Press

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