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Complexity of the Bollobás–Riordan Polynomial. Exceptional Points and Uniform Reductions

Research paper by Markus Bläser, Holger Dell, Johann A. Makowsky

Indexed on: 15 May '09Published on: 15 May '09Published in: Theory of Computing Systems



Abstract

The coloured Tutte polynomial by Bollobás and Riordan is, as a generalization of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contraction-deletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #P-hard to evaluate almost everywhere by establishing reductions along curves and lines.We establish a similar result for the coloured Tutte polynomial on integral domains. To capture the algebraic flavour and the uniformity inherent in this type of result, we introduce a new kind of reductions, uniform algebraic reductions, that are well-suited to investigate the evaluation complexity of graph polynomials. Our main result identifies a small, algebraic set of exceptional points and says that the evaluation problem of the coloured Tutte is equivalent for all non-exceptional points, under polynomial-time uniform algebraic reductions. On the way we obtain a self-contained proof for the difficult evaluations of the classical Tutte polynomial.