We present a new approximation algorithm for the (metric) prize-collecting traveling salesperson problem (PCTSP). In PCTSP, opposed to the classical traveling salesperson problem (TSP), one may not include a vertex of the input graph in the returned tour at the cost of a given vertex-dependent penalty, and the objective is to balance the length of the tour and the incurred penalties for omitted vertices by minimizing the sum of the two. We present an algorithm that achieves an approximation guarantee of $1.774$ with respect to the natural linear programming relaxation of the problem. This significantly reduces the gap between the approximability of classical TSP and PCTSP, beating the previously best known approximation factor of $1.915$. As a key ingredient of our improvement, we present a refined decomposition technique for solutions of the LP relaxation, and show how to leverage components of that decomposition as building blocks for our tours.

翻译：我们针对（度量）奖励收集旅行商问题（PCTSP）提出了一种新的近似算法。与经典旅行商问题（TSP）不同，在PCTSP中，允许在返回的环游中不包含输入图的某个顶点，但需支付给定的与顶点相关的惩罚成本，其目标是通过最小化环游长度与未访问顶点所产生的惩罚成本之和，来平衡两者。我们提出的算法相对于该问题的自然线性规划松弛，实现了$1.774$的近似保证。这显著缩小了经典TSP与PCTSP可近似性之间的差距，超越了此前已知的最佳近似因子$1.915$。作为我们改进的关键要素，我们提出了一种针对LP松弛解的精炼分解技术，并展示了如何利用该分解中的组件作为构建环游的模块。