An Algorithmic Framework for Locally Constrained Homomorphisms

TL;DR

This paper introduces a new algorithmic framework for analyzing the parameterized complexity of locally constrained graph homomorphism problems, providing new FPT, W[1]-hard, and para-NP-complete results.

Contribution

It develops a general ILP-based framework for establishing fixed-parameter tractability and hardness results for these problems, filling a gap in their complexity analysis.

Findings

Proves several new FPT results using the ILP framework.

Establishes W[1]-hardness and para-NP-completeness for various parameters.

Applies the framework to the Role Assignment problem from social network theory.

Abstract

A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively. The corresponding decision problems, LBHOM, LIHOM and LSHOM, are well studied both on general graphs and on special graph classes. Apart from complexity results when the problems are parameterized by the treewidth and maximum degree of the guest graph, the three problems still lack a thorough study of their parameterized complexity. This paper fills this gap: we prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph $G$. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.