On the hop-constrained Steiner tree problems
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The hop-constrained Steiner tree problem is a generalization of the classical Steiner tree problem, and asks for minimum cost subtree that spans some specified nodes of a given graph, such that the number of edges between each node of the tree and its root respects a given hop limit. This NP-hard problem has many variants, which are often modeled as integer linear programs. Two of the models are so called assignment and partial-ordering based models, which yield (up to our knowledge) the best two state-of-the-art formulations for the problem variant Steiner tree problem with revenues, budgets and hop constraints. We show that the linear programming relaxation of the partial-ordering based model is stronger than that of the assignment model for the hop-constrained Steiner tree problem; this remains true also for a class of hop-constrained problems, including the hop-constrained minimum spanning tree problem, the Steiner tree problem with revenues, budgets and hop constraints.
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