  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.
