Abstract
In this chapter we consider two combinatorial versions of the Steiner tree problem: the Steiner tree problem in graphs and the Steiner tree problem in hypergraphs. Also, we consider the minimum spanning tree problem in hypergraphs. Although this book focuses on geometric interconnection problems in the plane, these combinatorial problems are included for several reasons. Firstly, the Steiner tree problem in graphs is probably the best studied of all the many variants of the Steiner tree problem. Secondly, the fixed orientation Steiner tree problem in the plane (and specifically the rectilinear Steiner tree problem in the plane) can be reduced to the Steiner tree problem in graphs. Thirdly, the full Steiner tree concatenation phase of GeoSteiner, the most efficient exact algorithm for computing minimum Steiner trees in the plane, can be reduced to either the Steiner tree problem in graphs or the minimum spanning tree problem in hypergraphs.
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