Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory

Counting homomorphisms from a graph H into another graph G is a fundamental problem of (parameterized) counting complexity theory. In this work, we study the case where both graphs H and G stem from given classes of graphs: H is an element of H and G is an element of G. By this, we combine the structurally restricted version of this problem (where the class G = T is the set of all graphs), with the language-restricted version (where the class H = T is the set of all graphs). The structurally restricted version allows an exhaustive complexity classification for classes H : Either we can count all homomorphisms in polynomial time (if the treewidth of H is bounded), or the problem becomes #W[1]-hard [Dalmau, Jonsson, Th.Comp.Sci'04]. In contrast, in this work, we show that the combined view most likely does not admit such a complexity dichotomy. Our main result is a construction based on Kneser graphs that associates every problem P in #W[1] with two classes of graphs H and G such that the problem P is equivalent to the problem #HOM(H -> G) of counting homomorphisms from a graph in H to a graph in G. In view of Ladner's seminal work on the existence of NP-intermediate problems [J.ACM'75] and its adaptations to the parameterized setting, a classification of the class #W[1] in fixed-parameter tractable and #W[1]-complete cases is unlikely. Hence, obtaining a complete classification for the problem #HOM(H -> G) seems unlikely. Further, our proofs easily adapt to W[1] and the problem of deciding whether a homomorphism between graphs exists. In search of complexity dichotomies, we hence turn to special graph classes. Those classes include line graphs, claw-free graphs, perfect graphs, and combinations thereof, and F-colorable graphs for fixed graphs F. As a special case, we obtain an easy proof of the parameterized intractability result of the problem of counting k-matchings in bipartite graphs.