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]. 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) * poly(n), 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 Hom(H) problem cannot be solved in time (|V(H)| – ε)^tw(G) * poly(n), 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 from early 2000s. 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.

The talk is based joint work with Karolina Okrasa [SODA 2020]. The paper is here: https://epubs.siam.org/doi/abs/10.1137/1.9781611975994.97