On the Complexity of Some Facet-Defining Inequalities of the QAP-polytope
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The Quadratic Assignment Problem (QAP) is a well-known NP-hard problem that is equivalent to optimizing a linear objective function over the QAP polytope. The QAP polytope with parameter n - n - is defined as the convex hull of rank-1 matrices xx^T with x as the vectorized n× n permutation matrices. In this paper we consider all the known exponential-sized families of facet-defining inequalities of the QAP-polytope. We describe a new family of valid inequalities that we show to be facet-defining. We also show that membership testing (and hence optimizing) over some of the known classes of inequalities is coNP-complete. We complement our hardness results by showing a lower bound of 2^Ω(n) on the extension complexity of all relaxations of n for which any of the known classes of inequalities are valid.
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