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 \(L(v)\subseteq V(H)\) 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.