一维二部欧几里德旅行商问题的解
Solution for a bipartite Euclidean traveling-salesman problem in one dimension
摘要：
旅行商问题是组合优化问题中研究最多的问题之一，因为它的描述简单，求解困难。我们刻画了每个凸和增加的成本函数的最佳周期时，抛出的点是独立的，并具有相同的概率分布在一个紧凑的间隔。当距离函数是欧氏距离的平方时，我们计算每个点的平均最优成本。我们还表明，平均最优成本是不是一个自平均量，明确计算其分布在热力学极限的方差。此外，我们证明了最优周期的成本不小于相同的点集的最优分配的成本的两倍。有趣的是，这个束缚在热力学极限是饱和的。
英文摘要：
The traveling salesman problem is one of the most studied combinatorial optimization problems, because of the simplicity in its statement and the difficulty in its solution. We characterize the optimal cycle for every convex and increasing cost function when the points are thrown independently and with an identical probability distribution in a compact interval. We compute the average optimal cost for every number of points when the distance function is the square of the Euclidean distance. We also show that the average optimal cost is not a self-averaging quantity by explicitly computing the variance of its distribution in the thermodynamic limit. Moreover, we prove that the cost of the optimal cycle is not smaller than twice the cost of the optimal assignment of the same set of points. Interestingly, this bound is saturated in the thermodynamic limit.
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