# Affine maps between quadratic assignment polytopes and subgraph isomorphism polytopes

## TL;DR

This paper investigates the geometric relationship between two complex polytopes related to the quadratic assignment problem and subgraph isomorphism, establishing non-isomorphism and face/projection relations.

## Contribution

It proves that the quadratic assignment polytope and the Young polytope are not isomorphic and describes their face and projection relationships.

## Key findings

- QAP(n) and P((n-2,2)) are not isomorphic.
- QAP(n) is a face of P((2n-2,2)).
- P((n-2,2)) is a projection of QAP(n).

## Abstract

We consider two polytopes. The quadratic assignment polytope $QAP(n)$ is the convex hull of the set of tensors $x\otimes x$, $x \in P_n$, where $P_n$ is the set of $n\times n$ permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph $K_n$ we consider appropriate $\binom{n}{2} \times \binom{n}{2}$ permutation matrix of the edges of $K_n$. The Young polytope $P((n-2,2))$ is the convex hull of all such matrices. In 2009, S. Onn showed that the subgraph isomorphism problem can be reduced to optimization both over $QAP(n)$ and over $P((n-2,2))$. He also posed the question whether $QAP(n)$ and $P((n-2,2))$, having $n!$ vertices each, are isomorphic. We show that $QAP(n)$ and $P((n-2,2))$ are not isomorphic. Also, we show that $QAP(n)$ is a face of $P((2n-2,2))$, but $P((n-2,2))$ is a projection of $QAP(n)$.