Lower Bounds for the Graph Homomorphism Problem

Abstract

The graph homomorphism problem (HOM) asks whether the vertices of a given n-vertex graph G can be mapped to the vertices of a given h-vertex graph H such that each edge of G is mapped to an edge of H. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the 2-CSP problem. In this paper, we prove several lower bounds for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{\Omega \left( \frac{n \log h}{\log \log h}\right) }$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{\mathcal {O}(n\log {h})}$ is almost asymptotically tight.

We also investigate what properties of graphs G and H make it difficult to solve HOM(G, H). An easy observation is that an $\mathcal {O}(h^n)$ upper bound can be improved to $\mathcal {O}(h^{{\text {vc}}(G)})$ where ${\text {vc}}(G)$ is the minimum size of a vertex cover of G. The second lower bound $h^{\Omega ({\text {vc}}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the “right-hand side” graph H, it is known that HOM(G, H) can be solved in time $(f(\Delta (H)))^n$ and $(f({\text {tw}}(H)))^n$ where $\Delta (H)$ is the maximum degree of H and ${\text {tw}}(H)$ is the treewidth of H. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $\chi (H)$ does not exceed ${\text {tw}}(H)$ and $\Delta (H)+1$, it is natural to ask whether similar upper bounds with respect to $\chi (H)$ can be obtained. We provide a negative answer by establishing a lower bound $(f(\chi (H)))^n$ for every function f. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.

The full version of the paper is available at http://arxiv.org/abs/1502.05447