In this paper, we present a polynomial time 2-approximation algorithm for the {\em unrooted prize-collecting forest with $K$ components} (URPCF$_K$) problem, the goal of which is to find a forest with exactly $K$ connected components to minimize the weight of the forest plus the penalty incurred by the vertices not spanned by the forest. For its rooted version RPCF$_K$, a 2-approximation algorithm is known. For the unrooted version, transforming it into a rooted version by guessing roots runs in time exponentially depending on $K$, which is unacceptable when $K$ is not a constant. We conquer this challenge by designing a rootless growing plus rootless pruning algorithm. As an application, we make use of this algorithm to solve the {\em prize-collecting min-sensor sweep cover} problem, improving previous approximation ratio 8 to 5. Keywords: approximation algorithm, prize-collecting Steiner forest, sweep cover.

翻译：本文提出了一种针对“无根带罚金K分支森林”（URPCFₖ）问题的多项式时间2-近似算法，其目标是在找到一棵恰好包含K个连通分支的森林时，最小化森林权重与未被森林覆盖顶点所产生的罚金之和。对于其有根版本RPCFₖ，已知存在一个2-近似算法。而对于无根版本，通过猜测根节点将其转化为有根版本的方法，其运行时间对K呈指数依赖，当K非常数时不可接受。我们通过设计一种“无根生长+无根剪枝”算法攻克了这一挑战。作为应用，我们利用该算法求解“带罚金最小传感器扫描覆盖”问题，将近似比从8改进至5。关键词：近似算法，带罚金Steiner森林，扫描覆盖。