Counting List Homomorphisms from Graphs of Bounded Treewidth: Tight Complexity Bounds

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) subset of V (H) for every v is an element of V (G), a list homomorphism is a function f : V (G) -> V (H) that preserves the edges (i.e., uv is an element of E(G) implies f (u)f (v) is an element of E(H)) and respects the lists (i.e., f (v) is an element of 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 vertical bar V(H)vertical bar(t) center dot n(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 subset of 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