No abstract

We present several new techniques for dealing with the Steiner problem in (undirected) networks. We consider them as building blocks of an exact algorithm, but each of them could also be of interest in its own right. First, we consider some relaxations of integer programming formulations of this problem and investigate different methods for dealing with these relaxations, not only to obtain lower bounds, but also to get additional information which is used in the computation of upper bounds and in reduction techniques. Then, we modify some known reduction tests and introduce some new ones. We integrate some of these tests into a package with a small worst case time which achieves impressive reductions on a wide range of instances. On the side of upper bounds, we introduce the new concept of heuristic reductions. On the basis of this concept, we develop heuristics that achieve sharper upper bounds than the strongest known heuristics for this problem despite running times which are smaller by orders of magnitude. Finally, we integrate these blocks into an exact algorithm. We present computational results on a variety of benchmark instances. The results are clearly superior to those of all other exact algorithms known to the authors.

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 exc1usively, for computing lower bounds; but not much has been known ab out the relative quality of these relaxations. We compare all c1assical 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-c1ass relaxation (e.g. the multicommodity flow or the dicut relaxation) cannot be worse than the optimal value of a tree-c1ass 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.

No abstract

No abstract