2-Approximation for Prize-Collecting Steiner Forest

TL;DR

This paper presents a deterministic polynomial-time algorithm for the prize-collecting Steiner forest problem that achieves a 2-approximation, improving previous results and matching the best known approximation for Steiner forest.

Contribution

The paper introduces a new 2-approximation algorithm for PCSF, surpassing previous approximation factors and matching the Steiner forest problem's best known approximation.

Findings

Achieves a 2-approximation for PCSF

Surpasses the previous 2.54-approximation

Matches the Steiner forest problem's approximation factor

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.