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.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Averbouch, I., Makowsky, J.A.: The complexity of multivariate matching polynomials (January 2007) (preprint)
Bläser, M., Dell, H.: Complexity of the cover polynomial. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 801–812. Springer, Heidelberg (2007)
Bläser, M., Hoffmann, C.: On the complexity of the interlace polynomial. arXiv:0707.4565 (2007)
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and real computation. Springer, New York (1998)
Bollobás, B.: Modern Graph Theory. Springer, Heidelberg (1999)
Bollobás, B., Riordan, O.: A Tutte polynomial for coloured graphs. Combinatorics, Probability and Computing 8(1-2), 45–93 (1999)
Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. In: Grundlehren der mathematischen Wissenschaften, February 1997. Springer, Heidelberg (1997)
Dong, F.M., Koh, K.M., Teo, K.L.: Chromatic Polynomials and Chromaticity of Graphs. World Scientific, Singapore (2005)
Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial. In: Johnson, D.S., Feige, U. (eds.) STOC, pp. 459–468. ACM, New York (2007)
Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Mathematical Proceedings of the Cambridge Philosophical Society 108(1), 35–53 (1990)
Kauffman, L.H.: A Tutte polynomial for signed graphs. Discrete Applied Mathematics 25, 105–127 (1989)
Linial, N.: Hard enumeration problems in geometry and combinatorics. SIAM J. Algebraic Discrete Methods 7(2), 331–335 (1986)
Lotz, M., Makowsky, J.A.: On the algebraic complexity of some families of coloured Tutte polynomials. Advances in Applied Mathematics 32(1), 327–349 (2004)
Makowsky, J.A.: Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic 126, 1–3 (2004)
Makowsky, J.A.: From a zoo to a zoology: Towards a general theory of graph polynomials. Theory of Computing Systems (2008), ISSN 1432-4350
Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Webb, B.S. (ed.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 327, pp. 173–226. Cambridge University Press, Cambridge (2005)
Stanley, R.P.: Acyclic orientations of graphs. Discrete Mathematics 306(10-11), 905–909 (2006)
Welsh, D.J.A.: Matroid Theory. London Mathematical Society Monographs, vol. 8. Academic Press, London (1976)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bläser, M., Dell, H., Makowsky, J.A. (2008). Complexity of the Bollobás-Riordan Polynomial. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds) Computer Science – Theory and Applications. CSR 2008. Lecture Notes in Computer Science, vol 5010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79709-8_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-79709-8_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79708-1
Online ISBN: 978-3-540-79709-8
eBook Packages: Computer ScienceComputer Science (R0)