Tight Lower Bounds on Graph Embedding Problems

We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph G to graph H cannot be done in time |V(H)|o(|V(G)|). We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of |V(H)|o(|V(H)|)-time algorithm deciding if graph G is a subgraph of H. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems. Moreover, as a consequence of our reductions, conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.