  We present a new approximation algorithm for the (metric) prize-collecting
traveling salesperson problem (PCTSP). In PCTSP, opposed to the classical
traveling salesperson problem (TSP), one may not include a vertex of the input
graph in the returned tour at the cost of a given vertex-dependent penalty, and
the objective is to balance the length of the tour and the incurred penalties
for omitted vertices by minimizing the sum of the two. We present an algorithm
that achieves an approximation guarantee of $1.774$ with respect to the natural
linear programming relaxation of the problem. This significantly reduces the
gap between the approximability of classical TSP and PCTSP, beating the
previously best known approximation factor of $1.915$. As a key ingredient of
our improvement, we present a refined decomposition technique for solutions of
the LP relaxation, and show how to leverage components of that decomposition as
building blocks for our tours.
