Abstract
The Steiner forest problem asks for a minimum weight forest that spans a given number of terminal sets. We propose new cut- and flow-based integer linear programming formulations for the problem which yield stronger linear programming bounds than the two previous strongest formulations: The directed cut formulation (Balakrishnan et al. in Oper Res 37(5):716–740, 1989; Chopra and Rao in Math Prog 64(1):209–229, 1994) and the advanced flow formulation by Magnanti and Raghavan (Networks 45:61–79, 2005). We further introduce strengthening constraints and provide an example where the integrality gap of our models is 1.5. In an experimental evaluation, we show that the linear programming bounds of the new formulations are indeed strong on practical instances and that the related branch-and-cut algorithm outperforms algorithms based on the previous formulations.
Highlights
The Steiner forest problem (SFP) is one of the fundamental network design problems
Our contribution We propose two new formulations for the Steiner forest problem that combine the strong bounds of the improved flow formulation with the practical usefulness of the simpler cut models
Our new model is even stronger than the improved flow model by [25] and it is the strongest known model for the SFP
Summary
The Steiner forest problem (SFP) is one of the fundamental network design problems. Given an edge-weighted undirected graph G = (V , E) and K terminal sets T 1, . . . , T K ⊆ V , it asks for a minimum weight forest in G such that the nodes inside. When multiple sets are present, one directed tree per set is needed and these, in general, can impose conflicting orientations to the edges This is a major additional difficulty in solving the Steiner forest problem. Our contribution We propose two new formulations for the Steiner forest problem that combine the strong bounds of the improved flow formulation with the practical usefulness of the simpler cut models. Their corresponding LP relaxations are stronger than the improved flow relaxation by [25] and the directed cut relaxation, and as the undirected cut relaxation as well. We write Sk for the set of all cut-sets that are relevant for T k and S := S1 ∪ · · · ∪ SK for the set of all relevant cut-sets
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