跨越密集城市的流动推销员问题
Traveling salesman problem across dense cities
摘要：
考虑~ \(n\)节点~ \(\{X_i\}_{1 \leq i \leq n}\)独立地分布在~ \(N\)城市中，其中包含单位正方形~ \(S\)根据分布~ \(f.\)每个城市被建模为包含在~ \(S\)中的~ \(r_n \times r_n\)正方形，并且让~ \(TSPC_n\)表示包含所有~ \(n\)节点的最小长度循环的长度，对应于流动推销员问题（TSP）。我们得到了~ \(TSPC_n\)的方差估计，并证明了，如果城市是良好的连接和人口稠密在一定意义下，然后~ \(TSPC_n\)适当的中心和缩放收敛到零的概率。利用证明技巧，我们交替地得到了无约束情况下，当节点独立地分布在单位正方形\(TSPC_n.\)上时，最小长度圈的长度\(TSP_n\)的相应结果
英文摘要：
Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) distributed independently across~\(N\) cities contained with the unit square~\(S\) according to a distribution~\(f.\) Each city is modelled as an~\(r_n \times r_n\) square contained within~\(S\) and let~\(TSPC_n\) denote the length of the minimum length cycle containing all the~\(n\) nodes, corresponding to the traveling salesman problem (TSP). We obtain variance estimates for~\(TSPC_n\) and prove that if the cities are well-connected and densely populated in a certain sense, then~\(TSPC_n\) appropriately centred and scaled converges to zero in probability. We also obtain large deviation type estimates for~\(TSPC_n.\) Using the proof techniques, we alternately obtain corresponding results for the length~\(TSP_n\) of the minimum length cycle in the unconstrained case, when the nodes are independently distributed throughout the unit square~\(S.\)
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