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

Approximation Algorithm for Unrooted Prize-Collecting Forest with Multiple Components and Its Application on Prize-Collecting Sweep Coverage