A comparison of Steiner tree relaxations

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 1ow-class relaxation (e.g. the multicommodity 1ow 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 1ow-based relaxation, which is to the authors’ knowledge the strongest linear relaxation of polynomial size for the Steiner problem in networks. ? 2001 Elsevier Science B.V. All rights reserved.