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)


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.


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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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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