Fine-grained complexity of graph homomorphism problem for bounded-treewidth graphs
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For graphs G and H, a homomorphism from G to H is an edge-preserving mapping from the vertex set of G to the vertex set of H. For a fixed graph H, by Hom(H) we denote the computational problem which asks whether a given graph G admits a homomorphism to H. If H is a complete graph with k vertices, then Hom(H) is equivalent to the k-Coloring problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that Hom(H) is polynomial-time solvable if H is bipartite or has a vertex with a loop, and NP-complete otherwise [Hell and Nešetřil, JCTB 1990]. In this paper we are interested in the complexity of the problem, parameterized by the treewidth of the input graph G. If G has n vertices and is given along with its tree decomposition of width tw(G), then the problem can be solved in time |V(H)|^tw(G)· n^O(1), using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if H is a projective core, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the H problem cannot be solved in time (|V(H)|-ϵ)^tw(G)· n^O(1), for any ϵ > 0. This result provides a full complexity characterization for a large class of graphs H, as almost all graphs are projective cores. We also notice that the naive algorithm can be improved for some graphs H, and show a complexity classification for all graphs H, assuming two conjectures from algebraic graph theory. In particular, there are no known graphs H which are not covered by our result. In order to prove our results, we bring together some tools and techniques from algebra and from fine-grained complexity.
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