Tight Lower Bounds on Graph Embedding Problems

TL;DR

This paper establishes tight exponential lower bounds for fundamental graph embedding problems, showing that under ETH, no significantly faster algorithms than brute-force exist, thus closing the complexity gap.

Contribution

It provides the first tight lower bounds for graph homomorphism and subgraph isomorphism problems, matching brute-force algorithm complexities and extending to related problems.

Findings

Proves no ETH-based algorithm can decide graph homomorphism faster than exponential in |V(G)|

Shows a reduction from Graph Homomorphism to Subgraph Isomorphism, establishing similar lower bounds

Conditional lower bounds apply to related problems like Graph Minors and Quadratic Assignment.

Abstract

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.