Tight Lower Bounds on Graph Embedding Problems

• Published in: Journal of the ACM Vol. 64; no. 3; pp. 1 - 22
• Main Authors: Cygan, Marek, Fomin, Fedor V., Golovnev, Alexander, Kulikov, Alexander S., Mihajlin, Ivan, Pachocki, Jakub, Socała, Arkadiusz
• Format: Journal Article
• Language: English
• Published: New York Association for Computing Machinery 01.06.2017
• Subjects: Algorithms; Graph theory; Graphs; Isomorphism; Lower bounds; Operations research; Studies
• ISSN: 0004-5411; 1557-735X
• DOI: 10.1145/3051094

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.