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
bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main
result is a lower bound 2Ω(logloghnlogh).
This rules out the existence of a single-exponential algorithm and shows that
the trivial upper bound 2O(nlogh) 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 O(hn) upper
bound can be improved to O(hvc(G)) where
vc(G) is the minimum size of a vertex cover of G. The second
lower bound hΩ(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(Δ(H)))n and
(f(tw(H)))n where Δ(H) is the maximum degree of H
and 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 χ(H) does not exceed
tw(H) and Δ(H)+1, it is natural to ask whether similar
upper bounds with respect to χ(H) can be obtained. We provide a negative
answer to this question by establishing a lower bound (f(χ(H)))n for any
function f. We also observe that similar lower bounds can be obtained for
locally injective homomorphisms.Comment: 19 page