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Algorithmica

pp 1–16 | Cite as

Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

  • Cornelius Brand
  • Holger Dell
  • Marc Roth
Article

Abstract

Jaeger et al. (Math Proc Camb Philos Soc 108(1):35–53, 1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: the evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén (in: ICALP 2010, vol. 6198, pp. 426–437, Springer, Berlin, Heidelberg, 2010) and Husfeldt and Taslaman (in: IPEC 2010, vol. 6478, pp. 192–203, Springer, Berlin, Heidelberg, 2010) in combination with the results of Curticapean (in: ICALP 2015, pp. 380–392, Springer, 2015), extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line \(y=1\), which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time Open image in new window unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann (Inf Comput 125(1):1–12, 1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that the #P-hard cases cannot be solved in time Open image in new window unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time Open image in new window unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean and transfer it to systems of linear equations that might not directly correspond to interpolation.

Notes

Acknowledgements

We thank Radu Curticapean for various fruitful discussions and interactions, and also Leslie Ann Goldberg, Miki Hermann, Mark Jerrum, John Lapinskas, David Richerby, and all other participants of the “dichotomies” work group at the Simons Institute in the spring of 2016. Moreover, we thank Tyson Williams who in 2013 pointed the second author to [11], and Thore Husfeldt for encouraging us to pursue the publication of the conference version of this article.

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Authors and Affiliations

  1. 1.Saarland University and Cluster of Excellence (MMCI)SaarbrückenGermany

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