带约束的最小费用旅游问题
On the Constrained Least-cost Tour Problem
摘要：
我们引入约束最小成本旅游（CLT）问题：给定一个无向图的权重和成本函数的边缘，最小化总成本的旅游植根于一个开始顶点，使总重量位于给定的范围内。CLT与带利润的旅行推销员问题家族有关，但不同之处在于在边而不是顶点上定义权重函数，并且要求总权重在一个范围内而不是至少某个配额。我们证明了CLT是$\mathcal{NP}$ -困难的，即使在简单的情况下，当输入图是一个路径。我们推导出一个信息的下限放松的完整性的边缘，并提出了一个启发式的动机，这种放松。对于需要巡回赛是一个简单的周期的情况下，我们开发了两个算法，利用Suurballe的算法来找到低成本，重量可行的周期。我们通过解决一个影响城市人口的现实问题来展示我们的算法：寻找在伦敦市步行，跑步和骑自行车时最大限度地减少空气污染的路线。
英文摘要：
We introduce the Constrained Least-cost Tour (CLT) problem: given an undirected graph with weight and cost functions on the edges, minimise the total cost of a tour rooted at a start vertex such that the total weight lies within a given range. CLT is related to the family of Travelling Salesman Problems with Profits, but differs by defining the weight function on edges instead of vertices, and by requiring the total weight to be within a range instead of being at least some quota. We prove CLT is $\mathcal{NP}$-hard, even in the simple case when the input graph is a path. We derive an informative lower bound by relaxing the integrality of edges and propose a heuristic motivated by this relaxation. For the case that requires the tour to be a simple cycle, we develop two heuristics which exploit Suurballe's algorithm to find low-cost, weight-feasible cycles. We demonstrate our algorithms by addressing a real-world problem that affects urban populations: finding routes that minimise air pollution exposure for walking, running and cycling in the city of London.
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