Tight Bounds for Subgraph Isomorphism and Graph Homomorphism

We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $\mathrm{\mid }V(H){\mathrm{\mid }}^{o(\mathrm{\mid }V(G)\mathrm{\mid })}$. Combined with the reduction of Cygan, Pachocki, and Soca{\l}a, our result rules out (subject to ETH) a possibility of $\mathrm{\mid }V(G){\mathrm{\mid }}^{o(\mathrm{\mid }V(G)\mathrm{\mid })}$-time algorithm deciding if graph $H$ is a subgraph of $G$. 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.