A Computational Proof of Complexity of Some Restricted Counting Problems

  • Jin-Yi Cai
  • Pinyan Lu
  • Mingji Xia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5532)


We explore a computational approach to proving intractability of certain counting problems. More specifically we study the complexity of Holant of 3-regular graphs. These problems include concrete problems such as counting the number of vertex covers or independent sets for 3-regular graphs. The high level principle of our approach is algebraic, which provides sufficient conditions for interpolation to succeed. Another algebraic component is holographic reductions. We then analyze in detail polynomial maps on ℝ2 induced by some combinatorial constructions. These maps define sufficiently complicated dynamics of ℝ2 that we can only analyze them computationally. We use both numerical computation (as intuitive guidance) and symbolic computation (as proof theoretic verification) to derive that a certain collection of combinatorial constructions, in myriad combinations, fulfills the algebraic requirements of proving #P-hardness. The final result is a dichotomy theorem for a class of counting problems.


Vertex Cover Dichotomy Theorem Symbolic Computation Counting Problem Recursive Construction 
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  1. 1.
    Bulatov, A.A., Dalmau, V.: Towards a dichotomy theorem for the counting constraint satisfaction problem. Inf. Comput. 205(5), 651–678 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bulatov, A.A., Grohe, M.: The complexity of partition functions. Theor. Comput. Sci. 348(2-3), 148–186 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cai, J.-Y., Lu, P.: Holographic Algorithms: From Art to Science. In: Proceedings of STOC 2007, pp. 401–410 (2007)Google Scholar
  4. 4.
    Cai, J.-Y., Lu, P., Xia, M.: Holographic Algorithms by Fibonacci Gates and Holographic Reductions for Hardness. In: FOCS 2008, pp. 644–653 (2008)Google Scholar
  5. 5.
    Cai, J.-Y., Lu, P., Xia, M.: Holant Problems and Counting CSP. In: STOC 2009 (to appear, 2009)Google Scholar
  6. 6.
    Collins, G.: Quantifier Elimination for Real Closed Fields by Cylindric Algebraic Decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  7. 7.
    Creignou, N., Hermann, M.: Complexity of Generalized Satisfiability Counting Problems. Inf. Comput. 125(1), 1–12 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Creignou, N., Khanna, S., Sudan, M.: Complexity classifications of boolean constraint satisfaction problems. SIAM Monographs on Discrete Mathematics and Applications (2001)Google Scholar
  9. 9.
    Dyer, M.E., Goldberg, L.A., Jerrum, M.: The Complexity of Weighted Boolean #CSP CoRR abs/0704.3683 (2007)Google Scholar
  10. 10.
    Dyer, M.E., Goldberg, L.A., Paterson, M.: On counting homomorphisms to directed acyclic graphs. J. ACM 54(6) (2007)Google Scholar
  11. 11.
    Dyer, M.E., Greenhill, C.S.: The complexity of counting graph homomorphisms. Random Struct. Algorithms 17(3-4), 260–289 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Goldberg, L.A., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. CoRR abs/0804.1932 (2008)Google Scholar
  13. 13.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Combin. Theory Ser. B 48, 92–110 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Tarski, A.: A Decision Method for Elementary Algebra and Geometry, Univ. of Calif. (1951)Google Scholar
  15. 15.
    Vadhan, S.P.: The Complexity of Counting in Sparse, Regular, and Planar Graphs. SIAM J. Comput. 31(2), 398–427 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Valiant, L.G.: Holographic Algorithms (Extended Abstract). In: Proc. 45th IEEE Symposium on Foundations of Computer Science, pp. 306–315 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Pinyan Lu
    • 2
  • Mingji Xia
    • 1
    • 3
  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA
  2. 2.Microsoft Research AsiaBeijingP.R. China
  3. 3.State Key Laboratory of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingP.R. China

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