Approximation Algorithm for Unrooted Prize-Collecting Forest with Multiple Components and Its Application on Prize-Collecting Sweep Coverage
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In this paper, we present a polynomial time 2-approximation algorithm for the 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 prize-collecting min-sensor sweep cover problem, improving previous approximation ratio 8 to 5. Keywords: approximation algorithm, prize-collecting Steiner forest, sweep cover.
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