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)⊆V(H) for every v∈V(G), a {\em list homomorphism} is a function f:V(G)→V(H) that preserves the edges (i.e., uv∈E(G) implies f(u)f(v)∈E(H)) and respects the lists (i.e., f(v)∈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⋅nO(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) the following way: if H has a loop or is nonbipartite, then irr(H) is the maximum size of a set S⊆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. We show that, for every fixed graph H, the number of list homomorphisms from (G,L) to H * can be counted in time irr(H)t⋅nO(1) if a tree decomposition of G having width at most t is given in the input, and * cannot be counted in time (irr(H)−ϵ)t⋅nO(1) for any ϵ>0, even if a tree decomposition of G having width at most t is given in the input, unless the #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
2024-01-16
2024-04-22