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 (Formula Presented.). 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 “righthand 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 by establishing a lower bound (f(χ(H)))n for every function f. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.