Complexity of Correspondence Homomorphisms

Correspondence homomorphisms are both a generalization of standard homomorphisms and a generalization of correspondence colourings. For a fixed target graph H, the problem is to decide whether an input graph G, with each edge labeled by a pair of permutations of V (H), admits a homomorphism to H ’corresponding’ to the labels, in a sense explained below. We classify the complexity of this problem as a function of the fixed graph H. It turns out that there is dichotomy – each of the problems is polynomial-time solvable or NP-complete. While most graphs H yield NP-complete problems, there are interesting cases of graphs H for which we solve the problem by Gaussian elimination. We also classify the complexity of the analogous correspondence list homomorphism problems. In this note we only include the proofs for the case H is reflexive.