The graph homomorphism problem is a canonical $NP$-complete problem.
It generalizes various other well-studied problems such as graph
coloring and finding cliques. To get a better insight into a
combinatorial problem, one often studies relaxations of the problem.
We define fractional homomorphisms and pseudo-homomorphisms as natural
relaxations of graph homomorphisms.
In their paper Feige and Lovasz defined a semidefinite relaxation of
the homomorphism problem, which allowed them to obtain polynomial time
algorithms for certain special cases of the problem. Their relaxation
is defined in terms of the solution to a semidefinite program. Hence a
characterization of their relaxation in terms of known combinatorial
notions is desirable. We show that our pseudo-homomorphism is
equivalent to the relaxation defined by Feige and Lovasz.
Our definition of pseudo-homomorphism involves the classical theta
number first defined by Lovasz. Although general graph homomorphism
does not admit a simple forbidden subgraph characterization,
surprisingly we can show that there is a simple forbidden subgraph
characterization (the forbidden subgraph is a clique in this case) of
the fractional homomorphism. As a byproduct, we obtain a simpler proof
of the NP hardness of the fractional chromatic number, first proved
by Grotschel, Lovasz and Schrijver using the ellipsoid method. Finally, we briefly discuss how to apply these techniques to general
NP problems and describe a unified setting in which a wide variety of seemingly disparate polynomial time problems can be decided.