The Fine-Grained Complexity of Graph Homomorphism Parameterized by Clique-Width

TL;DR

This paper classifies the computational complexity of the graph homomorphism problem based on clique-width of the input graph, identifying precise bounds and properties of the target graph that determine solvability.

Contribution

It introduces the signature number of the target graph and establishes tight upper and lower bounds for the homomorphism problem based on clique-width, under the Strong Exponential Time Hypothesis.

Findings

Homomorphism problem solvable in O*(s(H)^{cw}) time

Matching lower bounds for projective core graphs

Classification covers all target graphs under conjectures

Abstract

The generic homomorphism problem, which asks whether an input graph $G$ admits a homomorphism into a fixed target graph $H$, has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of $G$ (denoted $\operatorname{cw}$) for virtually all choices of $H$ under the Strong Exponential Time Hypothesis. In particular, we identify a property of $H$ called the signature number $s(H)$ and show that for each $H$, the homomorphism problem can be solved in time $\mathcal{O}^*(s(H)^{\operatorname{cw}})$. Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each $H$ that is either a projective core or a graph admitting a factorization with additional properties -- allowing us to cover all possible target graphs under long-standing conjectures.