2-Approximation for Prize-Collecting Steiner Forest

• Published in: Journal of the ACM Vol. 72; no. 2; pp. 1 - 27
• Main Authors: Ahmadi, Ali, Gholami, Iman, Hajiaghayi, MohammadTaghi, Jabbarzade, Peyman, Mahdavi, Mohammad
• Format: Journal Article
• Language: English
• Published: New York, NY ACM 15.04.2025; Association for Computing Machinery
• Subjects: Algorithms; Approximation; Approximation algorithms; Graph algorithms; Iterative algorithms; Mathematics of computing; Polynomials; approximation algorithm; Steiner forest; prize-collecting
• ISSN: 0004-5411; 1557-735X
• DOI: 10.1145/3722551

Approximation algorithms for the prize-collecting Steiner forest (PCSF) problem have been a subject of research for more than three decades, starting with the seminal works of Agrawal et al. and Goemans and Williamson on Steiner forest and prize-collecting problems. In this article, 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, Könemann et al. 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 an iterative algorithm and a novel analysis technique. Since 2 is the best known approximation guarantee for the 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.