In this paper, we study the prize-collecting rural postman problem (PCRPP), a variant of the rural postman problem. Given a PCRPP instance consisting of an undirected graph whose edges have nonnegative lengths and nonnegative profits, together with a root vertex, the goal is to find a closed walk that starts and ends at the root vertex and minimizes the sum of its length and the profits of all edges that the walk does not traverse. A natural way to design an approximation algorithm for the PCRPP is to construct a prize-collecting traveling salesman problem (PCTSP) instance from the given PCRPP instance, apply an approximation algorithm to the PCTSP instance, and then convert the resulting PCTSP solution into a PCRPP solution. We show that this approach has an inherent factor-two barrier: even if the constructed PCTSP instance is solved exactly, the resulting PCRPP solution can have objective value arbitrarily close to twice the optimum value of the original PCRPP instance. Our main result is a polynomial-time approximation algorithm with approximation ratio strictly smaller than 1.6 for the PCRPP. On a public 118-instance benchmark set, the proposed algorithm has average and maximum optimality gaps of 3.39\% and 12.12\%, respectively.

翻译：本文研究了奖品收集农村邮递员问题(PCRPP)，这是农村邮递员问题的一个变种。给定一个PCRPP实例，包含一个无向图，其边具有非负长度和非负利润，以及一个根顶点，目标是找到一条从根顶点出发并返回根顶点的闭迹，该闭迹的长度与未被遍历的边的利润之和最小。设计PCRPP近似算法的一种自然方法是从给定的PCRPP实例构造一个奖品收集旅行商问题(PCTSP)实例，对PCTSP实例应用近似算法，然后将得到的PCTSP解转换为PCRPP解。我们表明，这种方法存在固有的因子二障碍：即使构造的PCTSP实例被精确求解，得到的PCRPP解的目标值也可能任意接近原始PCRPP实例最优值的两倍。我们的主要结果是一种多项式时间近似算法，其近似比严格小于1.6。在一个包含118个实例的公开基准测试集上，所提出算法的平均最优性差距为3.39%，最大最优性差距为12.12%。