Graph Homomorphisms with Complex Values: A Dichotomy Theorem

Abstract

Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function ZA(G) is defined as ZA(G) = σξ:V →[m] (u,v)∈E ΠAξ(u),ξ(v), where G = (V,E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of ZA(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.