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The complexity of graph homomorphism problems has been the subject of intense study. It is a long standing open problem to give a (decidable) complexity dichotomy theorem for the partition function of directed graph homomorphisms. In this paper, we prove a decidable complexity dichotomy theorem for this problem and our theorem applies to all non-negative weighted form of the problem: Given any fixed matrix A with non-negative entries, the partition function Z_A(G) of directed graph homomorphisms from any directed graph G is either tractable in polynomial time or #P-hard, depending on the matrix A. The proof of the dichotomy theorem is combinatorial, but involves the definition of an infinite family of graph homomorphism problems. The proof of its decidability is algebraic using properties of polynomials.

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