Theory of Computing Systems

, Volume 46, Issue 4, pp 690–706 | Cite as

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

Article

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.

Keywords

Computational complexity Tutte polynomial Graph polynomials Algebraic reductions 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Markus Bläser
    • 1
  • Holger Dell
    • 2
  • Johann A. Makowsky
    • 3
  1. 1.Saarland UniversitySaarbrückenGermany
  2. 2.Humboldt University of BerlinBerlinGermany
  3. 3.Technion-Israel Institute of TechnologyHaifaIsrael

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