The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs
G
,
H
, and lists
\(L(v)\subseteq V(H)\)
for every
\(v\in V(G)\)
, a
list homomorphism
is a function
\(f:V(G)\rightarrow V(H)\)
that preserves the edges (i.e.,
\(uv\in E(G)\)
implies
\(f(u)f(v)\in E(H)\)
) and respects the lists (i.e.,
\(f(v)\in L(v))\)
. Standard techniques show that if
G
is given with a tree decomposition of width
t
, then the number of list homomorphisms can be counted in time
\(|V(H)|^t\cdot n^{\mathcal {O}(1)}\)
. Our main result is determining, for every fixed graph
H
, how much the base
\(|V(H)|\)
in the running time can be improved. For a connected graph
H
, we define irr(
H
) in the following way: if
H
has a loop or is nonbipartite, then irr(
H
) is the maximum size of a set
\(S\subseteq V(H)\)
where any two vertices have different neighborhoods; if
H
is bipartite, then irr(
H
) is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected
H
, we define irr(
H
) as the maximum of irr(
C
) over every connected component
C
of
H
. It follows from earlier results that if irr(
H
)=1, then the problem of counting list homomorphisms to
H
is polynomial-time solvable, and otherwise it is #P-hard. We show that, for every fixed graph
H
, the number of list homomorphisms from
(G,L)
to
H
—
can be counted in time
\(\operatorname{irr}(H)^t\cdot n^{\mathcal {O}(1)}\)
if a tree decomposition of
G
having width at most
t
is given in the input, and,
—
given that
\(\operatorname{irr}(H)\ge 2\)
, cannot be counted in time
\((\operatorname{irr}(H)-\varepsilon)^t\cdot n^{\mathcal {O}(1)}\)
for any
\(\varepsilon \gt 0\)
, even if a tree decomposition of
G
having width at most
t
is given in the input, unless the Counting Strong Exponential-Time Hypothesis (#SETH) fails.
Thereby, we give a precise and complete complexity classification featuring matching upper and lower bounds for all target graphs with or without loops.