Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds

Abstract

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 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 \(|V(H)|^t\cdot n^{\mathcal {O}(1)} \). 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 \(\operatorname{irr}(H) \) in the following way: if H has a loop or is nonbipartite, then \(\operatorname{irr}(H) \) is the maximum size of a set S⊆V(H) where any two vertices have different neighborhoods; if H is bipartite, then \(\operatorname{irr}(H) \) is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected H, we define \(\operatorname{irr}(H) \) as the maximum of \(\operatorname{irr}(C) \) over every connected component C of H. It follows from earlier results that if \(\operatorname{irr}(H)=1 \), 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 \((\operatorname{irr}(H)-\epsilon)^t\cdot n^{\mathcal {O}(1)} \) for any ϵ > 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.