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 $n$ - \qappolytope{n} - is defined as the convex hull of rank-$1$ matrices $xx^T$ with $x$ as the vectorized $n\times n$ 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 $2^{\Omega(n)}$ on the extension complexity of all relaxations of \qappolytope{n} for which any of the known classes of inequalities are valid.

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关于 QAP-polytope 的某些方面定义不等式的复杂性

二次分配问题 (QAP) 是众所周知的 NP 难问题，相当于在 QAP 多胞体上优化线性目标函数。参数为 $n$ - \qappolytope{n} - 的 QAP polytope 被定义为 rank-$1$ 矩阵 $xx^T$ 的凸包，其中 $x$ 作为向量化的 $n\times n$ 置换矩阵。在本文中，我们考虑了 QAP 多胞体的所有已知指数大小的定义面不等式的族。我们描述了一组新的有效不等式，我们证明它们是定义方面的。我们还表明，对某些已知不等式类别的成员资格测试（以及因此优化）是 coNP 完全的。