Affine maps between quadratic assignment polytopes and subgraph isomorphism polytopes

We consider two polytopes. The quadratic assignment polytope QAP(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 appropriate (n 2 ) × (n 2 ) 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 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). The Boolean quadratic polytope is the convex hull of the set BQP(n) = {x⊗ x | x ∈ {0, 1}}, where x⊗x ∈ {0, 1} is the tensor product and n ∈ N, n ≥ 2. All polytopes considered in this paper are defined as convex hulls of sets of vertices. Therefore, we introduce the notation only for the set of vertices, and not for the whole polytope, and we often call the set of vertices a polytope. Recently, the properties of Boolean quadratic polytopes and of the affinely equivalent to them cut polytopes are studied quite intensively. In particular, this is confirmed by the large number of citations of the Deza and Laurent monograph [1], entirely devoted to this topic. It is known that BQP(n) is 3-neighborly [2] (every three vertices form a face of the polytope), it has an exponential complexity of an extended formulation [3, 5], it is affinely equivalent to some faces of polytopes associated with many other NP-hard problems (the traveling salesman problem, the knapsack problem, the set covering and set packing problems, the maximal 3-satisfiability problem, the 3-assignment problem, ∗Supported by the State Assignment for Research in P.G. Demidov Yaroslavl State University, 1.5768.2017/P220. The search engine scholar.google.com reports 895 citations on May 29, 2017