# On the family of 0/1-polytopes with NP-complete non-adjacency relation

## TL;DR

This paper explores the complexity of non-adjacency recognition in 0/1-polytopes, showing NP-completeness in certain families and the limitations of their relation to other NP-complete problems.

## Contribution

It demonstrates that the special family of 0/1-polytopes with NP-complete non-adjacency cannot be faces of polytopes from some well-known NP-complete problems, highlighting structural distinctions.

## Key findings

- Non-adjacency recognition is NP-complete for certain 0/1-polytopes.
- These polytopes are faces of polytopes related to several NP-complete problems.
- Such polytopes do not appear as faces in polytopes of maximum independent set, set packing, and 3-assignments.

## Abstract

In 1995 T. Matsui considered a special family 0/1-polytopes for which the problem of recognizing the non-adjacency of two arbitrary vertices is NP-complete. In 2012 the author of this paper established that all the polytopes of this family are present as faces in the polytopes associated with the following NP-complete problems: the traveling salesman problem, the 3-satisfiability problem, the knapsack problem, the set covering problem, the partial ordering problem, the cube subgraph problem, and some others. In particular, it follows that for these families the non-adjacency relation is also NP-complete. On the other hand, it is known that the vertex adjacency criterion is polynomial for polytopes of the following NP-complete problems: the maximum independent set problem, the set packing and the set partitioning problem, the three-index assignment problem. It is shown that none of the polytopes of the above-mentioned special family (with the exception of a one-dimensional segment) can be the face of polytopes associated with the problems of the maximum independent set, of a set packing and partitioning, and of 3-assignments.