The Dynamic Complexity of Acyclic Hypergraph Homomorphisms

TL;DR

This paper investigates the dynamic complexity of hypergraph homomorphism problems, demonstrating that certain cases can be maintained efficiently under updates, while others are unlikely to be manageable in the DynFO framework.

Contribution

It establishes the first dynamic maintenance results for acyclic hypergraph homomorphisms and highlights the limitations when both hypergraphs are updated.

Findings

Homomorphism existence can be maintained under single-edge updates for acyclic hypergraphs.

Maintaining homomorphisms becomes unlikely in DynFO when both hypergraphs are subject to updates.

The results delineate the boundary of dynamic tractability for hypergraph homomorphism problems.

Abstract

Finding a homomorphism from some hypergraph $\mathcal{Q}$ (or some relational structure) to another hypergraph $\mathcal{D}$ is a fundamental problem in computer science. We show that an answer to this problem can be maintained under single-edge changes of $\mathcal{Q}$, as long as it stays acyclic, in the DynFO framework of Patnaik and Immerman that uses updates expressed in first-order logic. If additionally also changes of $\mathcal{D}$ are allowed, we show that it is unlikely that existence of homomorphisms can be maintained in DynFO.