  Optimal assignment of classes to classrooms \cite{dickey}, design of DNA
microarrays \cite{carvalho}, cross species gene analysis \cite{kolar}, creation
of hospital layouts cite{elshafei}, and assignment of components to locations
on circuit boards \cite{steinberg} are a few of the many problems which have
been formulated as a quadratic assignment problem (QAP). Originally formulated
in 1957, the QAP is one of the most difficult of all combinatorial optimization
problems. Here, we use statistical mechanical methods to study the asymptotic
behavior of problems in which the entries of at least one of the two matrices
that specify the problem are chosen from a random distribution $P$.
Surprisingly, this case has not been studied before using statistical methods
despite the fact that the QAP was first proposed over 50 years ago
\cite{Koopmans}. We find simple forms for $C_{\rm min}$ and $C_{\rm max}$, the
costs of the minimal and maximum solutions respectively. Notable features of
our results are the symmetry of the results for $C_{\rm min}$ and $C_{\rm max}$
and the dependence on $P$ only through its mean and standard deviation,
independent of the details of $P$. After the asymptotic cost is determined for
a given QAP problem, one can straightforwardly calculate the asymptotic cost of
a QAP problem specified with a different random distribution $P$.
