The profitable tour problem (PTP) is a well-known NP-hard routing problem searching for a tour visiting a subset of customers while maximizing profit as the difference between total revenue collected and traveling costs. PTP is known to be solvable in polynomial time when special structures of the underlying graph are considered. However, the computational complexity of the corresponding probabilistic generalizations is still an open issue in many cases. In this paper, we analyze the probabilistic PTP where customers are located on a tree and need, with a known probability, for a service provision at a predefined prize. The problem objective is to select a priori a subset of customers with whom to commit the service so to maximize the expected profit. We provide a polynomial time algorithm computing the optimal solution in $O(n^2)$, where $n$ is the number of nodes in the tree.

翻译：利润丰厚的旅游问题(PTP)是一个众所周知的NP-硬路线问题,在寻找参观一组客户的旅游时,将利润最大化作为总收入和旅行费用之间的差额。在考虑基本图的特殊结构时,PTP在多元时间内是可溶的。然而,在很多情况下,相应的概率一般化的计算复杂性仍然是一个未决问题。在本文中,我们分析了顾客位于一棵树上的概率性PTP,在预先确定的奖励中需要服务。问题的目标是选择一个先期的客户子集,与这些客户一起承诺提供服务,以尽量扩大预期利润。我们用$(n%2)计算最佳解决办法,用$(n%2)计算,其中美元是树上的节点数。