Graph Homomorphisms with Complex Values: A Dichotomy Theorem

Abstract

Each symmetric matrix $\mathbf{A}$ over $\mathbb{C}$ defines a graph homomorphism function $Z_{\bf A}(\cdot)$ on undirected graphs. The function $Z_{\mathbf{A}} (\cdot)$ is also called the partition function from statistical physics, and can encode many interesting graph properties, including counting vertex covers and $k$-colorings. We study the computational complexity of $Z_{\mathbf{A}} (\cdot)$ for arbitrary symmetric matrices $\mathbf{A}$ with algebraic complex values. Building on work by Dyer and Greenhill [Random Structures and Algorithms, 17 (2000), pp. 260--289], Bulatov and Grohe [Theoretical Computer Science, 348 (2005), pp. 148--186], and especially the recent beautiful work by Goldberg et al. [SIAM J. Comput., 39 (2010), pp. 3336--3402], we prove a complete dichotomy theorem for this problem. We show that $Z_{\mathbf{A}} (\cdot)$ is either computable in polynomial-time or \#P-hard, depending explicitly on the matrix $\mathbf{A}$. We further prove that the tractability criterion on $\mathbf{A}$ is polynomial-time decidable.