  The prize-collecting Steiner tree problem PCSTP is a well-known
generalization of the classical Steiner tree problem in graphs, with a large
number of practical applications. It attracted particular interest during the
latest (11th) DIMACS Challenge and since then a number of PCSTP solvers have
been introduced in the literature, some of which drastically improved on the
best results achieved at the Challenge. The following article aims to further
advance the state of the art. It introduces new techniques and algorithms for
PCSTP, involving various forms of reductions of PCSTP instances to equivalent
problems---which for example allows to decrease the problem size or to obtain a
better IP formulation. Several of the new techniques and algorithms provably
dominate previous approaches. Further theoretical properties of the new
components, such as their complexity, are discussed, and their profound
interaction is described. Finally, the new developments also translate into a
strong computational performance: the resulting exact solver outperforms all
previous approaches---both in terms of run-time and solvability---and can solve
formerly intractable benchmark instances from the 11th DIMACS Challenge to
optimality.
