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Graph Homomorphisms with Complex Values: A Dichotomy Theorem

(Extended Abstract)
  • Jin-Yi Cai
  • Xi Chen
  • Pinyan Lu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

Abstract

Graph homomorphism problem has been studied intensively. Given an m ×m symmetric matrix A, the graph homomorphism function Z A (G) is defined as

$$ Z_{\rm A}(G) (G) = \sum_{\xi:V\rightarrow [m]} \prod_{(u,v)\in E} A_{\xi(u),\xi(v)}, $$

where G = (V, E) is any undirected graph. The function Z A (G) can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of Z A (G), for arbitrary complex valued symmetric matrices A. Building on the work by Dyer and Greenhill [1], Bulatov and Grohe [2], and especially the recent beautiful work by Goldberg, Grohe, Jerrum and Thurley [3], we prove a complete dichotomy theorem for this problem.

Keywords

Partition Function Undirected Graph Prime Power Constraint Satisfaction Problem Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Dyer, M., Greenhill, C.: The complexity of counting graph homomorphisms. In: Proceedings of the 9th International Conference on Random Structures and Algorithms, pp. 260–289 (2000)Google Scholar
  2. 2.
    Bulatov, A., Grohe, M.: The complexity of partition functions. Theoretical Computer Science 348(2), 148–186 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Goldberg, L., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. In: Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science, pp. 493–504 (2009)Google Scholar
  4. 4.
    Lovász, L.: Operations with structures. Acta Mathematica Hungarica 18, 321–328 (1967)zbMATHCrossRefGoogle Scholar
  5. 5.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)zbMATHCrossRefGoogle Scholar
  6. 6.
    Freedman, M., Lovász, L., Schrijver, A.: Reflection positivity, rank connectivity, and homomorphism of graphs. Journal of the American Mathematical Society 20, 37–51 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dyer, M., Goldberg, L., Paterson, M.: On counting homomorphisms to directed acyclic graphs. Journal of the ACM 54(6) (2007) Article 27Google Scholar
  8. 8.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. Journal of Combinatorial Theory, Series B 48(1), 92–110 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Thurley, M.: The complexity of partition functions on Hermitian matrices. arXiv (1004.0992) (2010)Google Scholar
  10. 10.
    Cai, J.Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: A dichotomy theorem. arXiv (0903.4728) (2009)Google Scholar
  11. 11.
    Schaefer, T.: The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pp. 216–226 (1978)Google Scholar
  12. 12.
    Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. In: SIAM Monographs on Discrete Mathematics and Applications (2001)Google Scholar
  13. 13.
    Feder, T., Vardi, M.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM Journal on Computing 28(1), 57–104 (1999)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Valiant, L.: Holographic algorithms. SIAM Journal on Computing 37(5), 1565–1594 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Valiant, L.: Accidental algorthims. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 509–517 (2006)Google Scholar
  16. 16.
    Cai, J.Y., Lu, P.: Holographic algorithms: from art to science. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 401–410 (2007)Google Scholar
  17. 17.
    Cai, J.Y., Lu, P., Xia, M.: Holant problems and counting CSP. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 715–724 (2009)Google Scholar
  18. 18.
    Feynman, R., Leighton, R., Sands, M.: The Feynman Lectures on Physics. Addison-Wesley, Reading (1970)Google Scholar
  19. 19.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998)Google Scholar
  20. 20.
    Ko, K.: Complexity Theory of Real Functions. Birkhäuser, Boston (1991)zbMATHGoogle Scholar
  21. 21.
    Lovász, L.: The rank of connection matrices and the dimension of graph algebras. European Journal of Combinatorics 27(6), 962–970 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lidl, R., Niederreiter, H.: Finite Fields. . Encyclopedia of Mathematics and its Applications, vol. 20. Cambridge University Press, Cambridge (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Xi Chen
    • 2
  • Pinyan Lu
    • 3
  1. 1.University of Wisconsin-Madison 
  2. 2.University of Southern California 
  3. 3.Microsoft Research Asia 

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