Fine-Grained Complexity of the Graph Homomorphism Problem for Bounded-Treewidth Graphs

For a fixed graph $H$, by Hom($H$) we denote the computational problem which asks whether given $G$ admits homomorphism to i.e., an edge-preserving mapping from $V(G)$ $V(H)$. As Hom($K_k$) is equivalent $k$-Coloring, homomorphisms can be seen as generalizations of colorings. It known that polynomial-time solvable if $H$ bipartite or has vertex with loop, and NP-complete otherwise [Hell Nešetřil, J. Comb. Theory Ser. B, 48 (1990), pp. 92--110]. In this paper are interested in complexity problem, parameterized treewidth input $G$. If $n$ vertices along its tree decomposition width ${tw}(G)$, then solved time $|V(H)|^{{tw}(G)} \cdot n^{\mathcal{O}(1)}$, using straightforward dynamic programming. We explore bound improved. show projective core, existence such faster algorithm unlikely: assuming Strong Exponential Time Hypothesis, cannot $(|V(H)|-\epsilon)^{{tw}(G)} for any $\epsilon > 0$. This result provides full characterization large class graphs almost all cores. also notice naive improved some classification two conjectures algebraic theory. particular, there no not covered our result.