In this paper, we introduce a polynomial-time 2-approximation algorithm for the Unrooted Prize-Collecting Forest with $K$ Components (URPCF$_K$) problem. URPCF$_K$ aims to find a forest with exactly $K$ connected components while minimizing both the forest's weight and the penalties incurred by unspanned vertices. Unlike the rooted version RPCF$_K$, where a 2-approximation algorithm exists, solving the unrooted version by guessing roots leads to exponential time complexity for non-constant $K$. To address this challenge, we propose a rootless growing and rootless pruning algorithm. We also apply this algorithm to improve the approximation ratio for the Prize-Collecting Min-Sensor Sweep Cover problem (PCMinSSC) from 8 to 5. Keywords: approximation algorithm, prize-collecting Steiner forest, sweep cover.

翻译：本文提出了一个针对无根$K$分量奖赏收集森林（URPCF$_K$）问题的多项式时间2-近似算法。URPCF$_K$旨在寻找一棵恰好包含$K$个连通分量的森林，同时最小化森林的权重与未被覆盖顶点产生的惩罚之和。与存在2-近似算法的有根版本RPCF$_K$不同，通过猜测根节点来求解无根版本会导致非常数$K$下的指数级时间复杂度。为解决这一挑战，我们提出了一种无根生长与无根修剪算法。我们还将该算法应用于奖赏收集最小传感器扫描覆盖问题（PCMinSSC），将其近似比从8改进至5。关键词：近似算法，奖赏收集斯坦纳森林，扫描覆盖。