Tight Bounds for Subgraph Isomorphism and Graph Homomorphism

TL;DR

This paper establishes tight exponential lower bounds for subgraph isomorphism and graph homomorphism problems, showing they cannot be solved faster than brute-force methods unless ETH fails, thus closing the complexity gap.

Contribution

The paper proves tight lower bounds for subgraph isomorphism and graph homomorphism problems, matching brute-force algorithm complexities under ETH, and closes the existing complexity gap.

Findings

Lower bounds match brute-force algorithms' running time

Deciding graph homomorphism cannot be done faster than exponential time

Results are conditioned on the Exponential Time Hypothesis (ETH)

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.