1 research outputs found
Counting and Finding Homomorphisms is Universal for Parameterized Complexity Theory
Counting homomorphisms from a graph into another graph is a
fundamental problem of (parameterized) counting complexity theory. In this
work, we study the case where \emph{both} graphs and stem from given
classes of graphs: and . 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 in with two classes of graphs
and such that the problem is
\emph{equivalent} to the problem of
counting homomorphisms from a graph in to a graph in
. In view of Ladner's seminal work on the existence of
-intermediate problems [J.ACM'75] and its adaptations to the
parameterized setting, a classification of the class in
fixed-parameter tractable and -complete cases is unlikely.
Hence, obtaining a complete classification for the problem seems unlikely. Further, our proofs easily
adapt to .
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 -colorable graphs for fixed graphs : If the
class is one of those classes and the class is
closed under taking minors, then we establish explicit criteria for the class
that partition the family of problems into polynomial-time solvable and
-hard cases. In particular, we can drop the condition of
being minor-closed for -colorable graphs.Comment: 42 pages, 8 figure