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Journal Article

A Comparison of Steiner Tree Relaxations


Polzin,  Tobias
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Polzin, T., & Vahdati Daneshmand, S. (2001). A Comparison of Steiner Tree Relaxations. Discrete Applied Mathematics, 112, 241-261.

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There are many (mixed) integer programming formulations of the Steiner problem in networks. The corresponding linear programming relaxations are of great interest particularly, but not exclusively, for computing lower bounds; but not much has been known about the relative quality of these relaxations. We compare all classical and some new relaxations from a theoretical point of view with respect to their optimal values. Among other things, we prove that the optimal value of a flow-class relaxation (e.g. the multicommodity flow or the dicut relaxation) cannot be worse than the optimal value of a tree-class relaxation (e.g. degree-constrained spanning tree relaxation) and that the ratio of the corresponding optimal values can be arbitrarily large. Furthermore, we present a new flow based relaxation, which is to the authors' knowledge the strongest linear relaxation of polynomial size for the Steiner problem in networks.