An Exact Algorithm for the Steiner Forest Problem

Authors Daniel R. Schmidt , Bernd Zey, François Margot

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Author Details

Daniel R. Schmidt
  • Institut für Informatik, Universität zu Köln, Germany
Bernd Zey
  • Fakultät für Informatik, TU Dortmund, Germany
François Margot
  • Carnegie-Mellon-University, Pittsburgh PA, USA

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Daniel R. Schmidt, Bernd Zey, and François Margot. An Exact Algorithm for the Steiner Forest Problem. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 70:1-70:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


The Steiner forest problem asks for a minimum weight forest that spans a given number of terminal sets. The problem has famous linear programming based 2-approximations [Agrawal et al., 1995; Goemans and Williamson, 1995; Jain, 2001] whose bottleneck is the fact that the most natural formulation of the problem as an integer linear program (ILP) has an integrality gap of 2. We propose new cut-based ILP formulations for the problem along with exact branch-and-bound based algorithms. While our new formulations cannot improve the integrality gap, we can prove that one of them yields stronger linear programming bounds than the two previous strongest formulations: The directed cut formulation [Balakrishnan et al., 1989; Chopra and Rao, 1994] and the advanced flow-based formulation by Magnanti and Raghavan [Magnanti and Raghavan, 2005]. In an experimental evaluation, we show that the linear programming bounds of the new formulations are indeed strong on practical instances and that our new branch-and-bound algorithms outperform branch-and-bound algorithms based on the previous formulations. Our formulations can be seen as a cut-based analogon to [Magnanti and Raghavan, 2005], whose existence was an open problem.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • branch-and-bound algorithms
  • Steiner network problems


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