随机二次指派问题
The Random Quadratic Assignment Problem
摘要：
课堂的最佳分配\cite{dickey}，DNA微阵列\cite{carvalho}的设计，跨物种基因分析\cite{kolar}，创建医院布局引用{elshafei}，以及分配元件到电路板上的位置\cite{steinberg}是已被公式化为二次分配问题（QAP）的许多问题中的几个。最初制定于1957年，QAP是所有组合优化问题中最困难的问题之一。在这里，我们使用统计力学的方法来研究的渐近行为的问题，其中至少有一个的两个矩阵，指定的问题是从一个随机分布$P$选择的条目。令人惊讶的是，这种情况下没有研究之前，使用统计方法，尽管事实上，QAP首次提出超过50年前\cite{Koopmans}。我们找到简单的形式为$C_{\rm min}$和$C_{\rm max}$，成本的最小和最大的解决方案。值得注意的特点，我们的研究结果是对称的结果为$C_{\rm min}$和$C_{\rm max}$和依赖于$P$仅通过其平均值和标准差，独立的细节$P$。在确定给定QAP问题的渐近成本之后，可以直接计算用不同随机分布$P$指定的QAP问题的渐近成本。
英文摘要：
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 $P$.
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