Affine maps between quadratic assignment polytopes and subgraph isomorphism polytopes

Abstract

We consider two polytopes. The quadratic assignment polytope PQAP (n) is the convex hull of the set of tensors x ⊗ x, x ∈ Pn , where Pn is the set of n × n permutation matrices. The second polytope is defined as follows. For every permutation of vertices of the complete graph Kn we consider the appropriate permutation matrix of the edges of Kn . 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 PQAP (n) and over P((n – 2, 2)). He also posed the question whether PQAP (n) and P((n – 2, 2)), having n! vertices each, are isomorphic. We show that PQAP (n) and P((n – 2, 2)) are not isomorphic. Also, we show that PQAP (n) is a face of P((2n 2 – 2, 2)), but P((n – 2, 2)) is a projection of PQAP (n).