The prize-collecting Steiner tree problem (PCSTP) is a well-known generalization of the classic Steiner tree problem in graphs, with a large number of practical applications. It attracted particular interest during the 11th DIMACS Challenge in 2014, and since then, several PCSTP solvers have been introduced in the literature. Although these new solvers further, and often drastically, improved on the results of the DIMACS Challenge, many PCSTP benchmark instances have remained unsolved. The following article describes further advances in the state of the art in exact PCSTP solving. It introduces new techniques and algorithms for PCSTP, involving various new transformations (or reductions) of PCSTP instances to equivalent problems, for example, to decrease the problem size or to obtain a better integer programming 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. Also, new complexity results for the exact solution of PCSTP and related problems are described, which form the base of the algorithm design. Finally, the new developments also translate into a strong computational performance: the resulting exact PCSTP solver outperforms all previous approaches, both in terms of runtime and solvability. In particular, it solves several formerly intractable benchmark instances from the 11th DIMACS Challenge to optimality. Moreover, several recently introduced large-scale instances with up to 10 million edges, previously considered to be too large for any exact approach, can now be solved to optimality in less than two hours. Summary of Contribution: The prize-collecting Steiner tree problem (PCSTP) is a well-known generalization of the classic Steiner tree problem in graphs, with many practical applications. The article introduces and analyses new techniques and algorithms for PCSTP that ultimately aim for improved (practical) exact solution. The algorithmic developments are underpinned by results on theoretical aspects, such as fixed-parameter tractability of PCSTP. Computationally, we considerably push the limits of tractibility, being able to solve PCSTP instances with up to 10 million edges. The new solver, which also considerably outperforms the state of the art on smaller instances, will be made publicly available as part of the SCIP Optimization Suite.
Near-linear Time Approximation Schemes for Clustering in Doubling Metrics
