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Journal Article

#### A Comparison of Steiner Tree Relaxations

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##### Citation

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

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-3158-8

##### Abstract

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.