Generalisations of Matrix Partitions : Complexity and Obstructions

A trigraph is a graph where each pair of vertices is labelled either 0 (a non-edge), 1 (an edge) or $\star$ (both an edge and a non-edge). In a series of papers, Hell et al. proposed to study the complexity of homomorphisms from graphs to trigraphs, called Matrix Partition Problems, where edges and non-edges can be both mapped to $\star$-edges, while a non-edge cannot be mapped to an edge, and vice-versa. Even though, Matrix Partition Problems are generalisations of CSPs, they share with them the property of being "intrinsically" combinatorial. So, the question of a possible dichotomy, i.e. P-time vs NP-complete, is a very natural one and raised in Hell et al.'s papers. We propose in this paper to study Matrix Partition Problems on relational structures, wrt a dichotomy question, and, in particular, homomorphisms between trigraphs. We first show that trigraph homomorphisms and Matrix Partition Problems are P-time equivalent, and then prove that one can also restrict (wrt dichotomy) to relational structures with one relation. Failing in proving that Matrix Partition Problems on directed graphs are not P-time equivalent to Matrix Partitions on relational structures, we give some evidence that it is unlikely by showing that reductions used in the case of CSPs cannot work. We turn then our attention to Matrix Partitions with finite sets of obstructions. We show that, for a fixed trigraph, the set of inclusion-wise minimal obstructions is finite for directed graphs if and only if it is finite for trigraphs. We also prove similar results for relational structures. We conclude by showing that on trees (seen as trigraphs) it is NP-complete to decide whether a given tree has a trigraph homomorphism to another input trigraph. The latter shows a notable difference on tractability between CSP and Matrix Partition as it is well-known that CSP is tractable on the class of trees.