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arXiv · 2309.05172

2-Approximation for Prize-Collecting Steiner Forest

Ali Ahmadi·Iman Gholami·MohammadTaghi Hajiaghayi·Peyman Jabbarzade·Mohammad Mahdavi
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Abstract

Approximation algorithms for the prize-collecting Steiner forest problem (PCSF) have been a subject of research for over three decades, starting with the seminal works of Agrawal, Klein, and Ravi and Goemans and Williamson on Steiner forest and prize-collecting problems. In this paper, we propose and analyze a natural deterministic algorithm for PCSF that achieves a $2$-approximate solution in polynomial time. This represents a significant improvement compared to the previously best known algorithm with a $2.54$-approximation factor developed by Hajiaghayi and Jain in 2006. Furthermore, K{\"{o}}nemann, Olver, Pashkovich, Ravi, Swamy, and Vygen have established an integrality gap of at least $9/4$ for the natural LP relaxation for PCSF. However, we surpass this gap through the utilization of a combinatorial algorithm and a novel analysis technique. Since $2$ is the best known approximation guarantee for Steiner forest problem, which is a special case of PCSF, our result matches this factor and closes the gap between the Steiner forest problem and its generalized version, PCSF.

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Ali Ahmadi, Iman Gholami, MohammadTaghi Hajiaghayi, Peyman Jabbarzade, Mohammad Mahdavi. 2023-09-11. 2-Approximation for Prize-Collecting Steiner Forest. https://arxiv.org/abs/2309.05172

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