Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory
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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∈H and G∈G. By this, we combine the structurally restricted version of this problem, with the language-restricted version. 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]. 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: If the class G is one of those classes and the class H is closed under taking minors, then we establish explicit criteria for the class H that partition the family of problems # HOM(H→G) into polynomial-time solvable and #W[1]-hard cases. In particular, we can drop the condition of H being minor-closed for F-colorable graphs.
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