Weak embeddings of posets to the Boolean lattice

TL;DR

This paper proves that determining weak embeddings of posets into small Boolean lattices or specific levels is NP-complete, and also shows the NP-completeness of embedding graphs into the middle levels of hypercubes.

Contribution

It establishes NP-completeness results for several poset embedding problems and related graph embedding problems, answering longstanding open questions.

Findings

NP-completeness of embedding posets into small Boolean lattices

NP-completeness of embedding posets into specific levels of Boolean lattices

NP-completeness of embedding graphs into the middle levels of hypercubes

Abstract

The goal of this paper is to prove that several variants of deciding whether a poset can be (weakly) embedded into a small Boolean lattice, or to a few consecutive levels of a Boolean lattice, are NP-complete, answering a question of Griggs and of Patkos. As an equivalent reformulation of one of these problems, we also derive that it is NP-complete to decide whether a given graph can be embedded to the two middle levels of some hypercube.