Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem:
$\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$,
where $S_n$ denotes the collection of all permutations on $x$.
I wonder if there is any sufficient condition on the matrix $Q$ that guarantees that the solution is given by the permutation that puts $x$ in increasing order. For instance, such a $Q$ qualifies: $Q = \left(\begin{matrix} 0 & -1 & 0 & 0 \\\ -1 & 0 & 0 & 0 \\\ 0 & 0 & 0 & -1 \\\ 0 & 0 & -1 & 0\end{matrix}\right)$. Does it have anything to do with majorization theory?
P.S., I googled a bit and it seems this is a particularization of the so-called quadratic assignment problem. There are a lot of discussions dealing with complexity of finding the solution. I wonder if we can we come up with some sufficient condition to guarantee that the optimal solution is to simply perform a sort.
