Solving the bi-objective prize-collecting Steiner tree problem with the ǫ-constraint method Markus Leitner a,1 Ivana Ljubi´ c b,2 Markus Sinnl b,3 a Institute of Computer Graphics and Algorithms, Vienna University of Technology, Vienna, Austria b Department of Statistics and Operations Research, University of Vienna, Vienna, Austria Abstract In this paper, we study the bi-objective prize-collecting Steiner tree problem, whose goal is to find a subtree that minimizes the edge costs for building that tree, and, at the same time, to maximize the collected node revenues. We propose to solve the problem using an ǫ-constraint algorithm. This is an iterative mixed-integer- programming framework that identifies one solution for every point on the Pareto front. In this framework, a branch-and-cut approach for the single-objective variant of the problem is enhanced with warm-start procedures that are used to (i) generate feasible solutions, (ii) generate violated cutting planes, and (iii) guide the branching process. Standard benchmark instances from the literature are used to assess the efficacy of our method. Keywords: bi-objective combinatorial optimization, Steiner tree problem, ǫ-constraint method 1 Supported by the Austrian Science Fund (FWF) under grant I892-N23. Email: leitner@ads.tuwien.ac.at 2 Supported by the APART Fellowship of the Austrian Academy of Sciences. Email: ivana.ljubic@univie.ac.at 3 Email: markus.sinnl@univie.ac.at Available online at www.sciencedirect.com Electronic Notes in Discrete Mathematics 41 (2013) 181–188 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. www.elsevier.com/locate/endm http://dx.doi.org/10.1016/j.endm.2013.05.091