Finegrained complexity of the graph homomorphism problem for boundedtreewidth graphs
Abstract
For graphs $G$ and $H$, a \emph{homomorphism} from $G$ to $H$ is an edgepreserving mapping from the vertex set of $G$ to the vertex set of $H$. For a fixed graph $H$, by \textsc{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 \textsc{Hom($H$)} is equivalent to the $k$\textsc{Coloring} problem, so graph homomorphisms can be seen as generalizations of colorings. It is known that \textsc{Hom($H$)} is polynomialtime solvable if $H$ is bipartite or has a vertex with a loop, and NPcomplete 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 $\mathrm{tw}(G)$, then the problem can be solved in time $V(H)^{\mathrm{tw}(G)} \cdot n^{\mathcal{O}(1)}$, using a straightforward dynamic programming. We explore whether this bound can be improved. We show that if $H$ is a \emph{projective core}, then the existence of such a faster algorithm is unlikely: assuming the Strong Exponential Time Hypothesis (SETH), the \textsc{Hom($H$)} problem cannot be solved in time $(V(H)\epsilon)^{\mathrm{tw}(G)} \cdot n^{\mathcal{O}(1)}$, for any $\epsilon > 0$. 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 finegrained complexity.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.08371
 Bibcode:
 2019arXiv190608371O
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Discrete Mathematics;
 Computer Science  Data Structures and Algorithms
 EPrint:
 An extended abstract of this paper appeared on SODA 2020