Prize-Collecting Steiner Tree (PCST) is a generalization of the Steiner Tree problem, a fundamental problem in computer science. In the classic Steiner Tree problem, we aim to connect a set of vertices known as terminals using the minimum-weight tree in a given weighted graph. In this generalized version, each vertex has a penalty, and there is flexibility to decide whether to connect each vertex or pay its associated penalty, making the problem more realistic and practical. Both the Steiner Tree problem and its Prize-Collecting version had long-standing $2$-approximation algorithms, matching the integrality gap of the natural LP formulations for both. This barrier for both problems has been surpassed, with algorithms achieving approximation factors below $2$. While research on the Steiner Tree problem has led to a series of reductions in the approximation ratio below $2$, culminating in a $\ln(4)+\epsilon$ approximation by Byrka, Grandoni, Rothvo{\ss}, and Sanit\`a, the Prize-Collecting version has not seen improvements in the past 15 years since the work of Archer, Bateni, Hajiaghayi, and Karloff, which reduced the approximation factor for this problem from $2$ to $1.9672$. Interestingly, even the Prize-Collecting TSP approximation, which was first improved below $2$ in the same paper, has seen several advancements since then. In this paper, we reduce the approximation factor for the PCST problem substantially to 1.7994 via a novel iterative approach.

翻译：奖项收集斯坦纳树（Prize-Collecting Steiner Tree, PCST）是计算机科学基础问题——斯坦纳树问题的一种推广。在经典斯坦纳树问题中，我们需要在给定加权图中用最小权重树连接一组称为终端的顶点。在此推广版本中，每个顶点都有罚金，且可灵活决定是连接每个顶点还是支付其关联罚金，这使得问题更具现实性和实用性。斯坦纳树问题及其奖项收集版本长期存在匹配各自自然线性规划间隙的$2$近似算法。这两个问题的这一障碍已被突破，实现了小于$2$的近似因子。尽管斯坦纳树问题的研究已将其近似比逐步降至$2$以下，最终由Byrka、Grandoni、Rothvo{\ss}和Sanit\`a实现了$\ln(4)+\epsilon$近似，但奖项收集版本自15年前Archer、Bateni、Hajiaghayi和Karloff的工作（将该问题近似因子从$2$降至$1.9672$）以来未见改进。有趣的是，同一篇论文中首次将奖项收集旅行商问题（Prize-Collecting TSP）近似比降至$2$以下后，该问题此后已取得多项进展。本文通过一种新颖的迭代方法，将PCST问题的近似因子大幅降低至1.7994。