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arXiv · 1507.03738

Tight Bounds for Subgraph Isomorphism and Graph Homomorphism

Fedor V. Fomin·Alexander Golovnev·Alexander S. Kulikov·Ivan Mihajlin
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Abstract

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 $|V(H)|^{o(|V(G)|)}$. Combined with the reduction of Cygan, Pachocki, and Soca{\l}a, our result rules out (subject to ETH) a possibility of $|V(G)|^{o(|V(G)|)}$-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.

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Fedor V. Fomin, Alexander Golovnev, Alexander S. Kulikov, Ivan Mihajlin. 2015-07-14. Tight Bounds for Subgraph Isomorphism and Graph Homomorphism. https://arxiv.org/abs/1507.03738

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