Abstract
A dichotomy theorem for counting problems due to Creignou and Hermann states that or any finite set S of logical relations, the counting problem #SAT(S) is either in FP, or #P-complete. In the present paper we show a dichotomy theorem for polynomial evaluation. That is, we show that for a given set S, either there exists a VNP-complete family of polynomials associated to S, or the associated families of polynomials are all in VP. We give a concise characterization of the sets S that give rise to “easy” and “hard” polynomials. We also prove that several problems which were known to be # P-complete under Turing reductions only are in fact # P-complete under many-one reductions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bürgisser, P.: On the structure of Valiant’s complexity classes. Discrete Mathematics and Theoretical Computer Science 3, 73–94 (1999)
Bürgisser, P.: Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics, vol. 7. Springer, Heidelberg (2000)
Briquel, I., Koiran, P.: A dichotomy theorem for polynomial evaluation, http://prunel.ccsd.cnrs.fr/ensl-00360974
Bulatov, A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. Journal of the ACM 53(1), 66–120 (2006)
Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Information and Computation 125, 1–12 (1996)
Creignou, N., Khanna, S., Sudan, M.: Complexity classification of boolean constraint satisfaction problems. SIAM monographs on discrete mathematics (2001)
Dong, F.M., Hendy, M.D., Teo, K.L., Little, C.H.C.: The vertex-cover polynomial of a graph. Discrete Mathematics 250(1-3), 71–78 (2002)
Jerrum, M.: Two-dimensional monomer-dimer systems are computationally intractable. Journal of Statistical Physics 48, 121–134 (1987)
Jerrum, M.: Counting, Sampling and Integrating: Algorithms and Complexity. Lectures in Mathematics - ETH Zürich. Birkhäuser, Basel (2003)
Linial, N.: Hard enumeration problems in geometry and combinatorics. SIAM Journal of Algebraic and 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)
Provan, J.S., Ball, M.O.: The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. of Comp. 12(4), 777–788 (1983)
Schaefer, T.J.: The complexity of satisfiability problems. In: Conference Record of the 10th Symposium on Theory of Computing, pp. 216–226 (1978)
Valiant, L.G.: Completeness classes in algebra. In: Proc. 11th ACM Symposium on Theory of Computing, pp. 249–261 (1979)
Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8, 189–201 (1979)
Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal of Computing 8(3), 410–421 (1979)
Zankó, V.: #P-completeness via many-one reductions. International Journal of Foundations of Computer Science 2(1), 77–82 (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Briquel, I., Koiran, P. (2009). A Dichotomy Theorem for Polynomial Evaluation. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-03816-7_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03815-0
Online ISBN: 978-3-642-03816-7
eBook Packages: Computer ScienceComputer Science (R0)