# 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)⊆ 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· 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) 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· n^𝒪(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· n^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 #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.

• 9 publications
• 44 publications
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06/19/2020

### Full complexity classification of the list homomorphism problem for bounded-treewidth graphs

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### List homomorphism problems for signed graphs

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### List homomorphisms by deleting edges and vertices: tight complexity bounds for bounded-treewidth graphs

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### Component twin-width as a parameter for BINARY-CSP and its semiring generalisations

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### Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

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### Parameterized Lower Bounds for Problems in P via Fine-Grained Cross-Compositions

We provide a general framework to exclude parameterized running times of...
09/24/2020

### Fine-grained complexity of the list homomorphism problem: feedback vertex set and cutwidth

For graphs G,H, a homomorphism from G to H is an edge-preserving mapping...