  Prize-Collecting Steiner Tree (PCST) is a generalization of the Steiner Tree
problem, a fundamental problem in computer science. In the classic Steiner Tree
problem, we aim to connect a set of vertices known as terminals using the
minimum-weight tree in a given weighted graph. In this generalized version,
each vertex has a penalty, and there is flexibility to decide whether to
connect each vertex or pay its associated penalty, making the problem more
realistic and practical.
  Both the Steiner Tree problem and its Prize-Collecting version had
long-standing $2$-approximation algorithms, matching the integrality gap of the
natural LP formulations for both. This barrier for both problems has been
surpassed, with algorithms achieving approximation factors below $2$. While
research on the Steiner Tree problem has led to a series of reductions in the
approximation ratio below $2$, culminating in a $\ln(4)+\epsilon$ approximation
by Byrka, Grandoni, Rothvo{\ss}, and Sanit\`a, the Prize-Collecting version has
not seen improvements in the past 15 years since the work of Archer, Bateni,
Hajiaghayi, and Karloff, which reduced the approximation factor for this
problem from $2$ to $1.9672$. Interestingly, even the Prize-Collecting TSP
approximation, which was first improved below $2$ in the same paper, has seen
several advancements since then.
  In this paper, we reduce the approximation factor for the PCST problem
substantially to 1.7994 via a novel iterative approach.
