二维二部旅行商和二因素问题平均最优费用的精确值
Exact value for the average optimal cost of bipartite traveling-salesman and 2-factor problems in two dimensions
摘要：
我们表明，平均成本的旅行推销员问题在两个维度，这是组合优化中的原型问题，在二分的情况下，是简单的分配问题的平均成本与相同的欧几里得，增加，凸权重。用这种方法，我们推广了一个已知的结果在一维精确解是availaible。因此，当成本函数是点之间的距离的平方时，最近确定的分配的平均成本为大量点$2N$提供了精确的预测$$\overline{E_N} = \frac{1}π\, \log N$$。作为我们分析的副产品，环路覆盖问题也具有相同的最优平均成本。我们还解释了为什么这个结果不能在更高的维度扩展。我们用数字来检验准确的预测。
英文摘要：
We show that the average cost for the traveling-salesman problem in two dimensions, which is the archetypal problem in combinatorial optimization, in the bipartite case, is simply related to the average cost of the assignment problem with the same Euclidean, increasing, convex weights. In this way we extend a result already known in one dimension where exact solutions are avalaible. The recently determined average cost for the assignment when the cost function is the square of the distance between the points provides therefore an exact prediction $$\overline{E_N} = \frac{1}π\, \log N$$ for large number of points $2N$. As a byproduct of our analysis also the loop covering problem has the same optimal average cost. We also explain why this result cannot be extended at higher dimensions. We numerically check the exact predictions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
