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

TL;DR

This paper presents a polynomial-time 2-approximation algorithm for the unrooted prize-collecting forest problem with multiple components, improving solutions for related sweep coverage problems.

Contribution

It introduces a novel rootless growing and pruning algorithm for the unrooted prize-collecting forest problem, reducing complexity and enhancing approximation ratios.

Findings

Developed a 2-approximation algorithm for URPCF_K.

Improved the approximation ratio for PCMinSSC from 8 to 5.

Provided a polynomial-time solution for a previously complex problem.

Abstract

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.