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Steiner Trees in Graphs and Hypergraphs

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Optimal Interconnection Trees in the Plane

Part of the book series: Algorithms and Combinatorics ((AC,volume 29))

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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Notes

  1. 1.

    The Steiner tree problem in graphs was originally formulated by Hakimi [185] (and independently by Levin [251]) in 1971. In the literature the problem is sometimes called the Steiner problem in networks (and the graph version is reserved for the unweighted case). Exact algorithms based on enumeration and dynamic programming were first proposed by Hakimi [185], Levin [251] and Dreyfus and Wagner [131].

  2. 2.

    For hypergraphs with edges of size 3, there exists a polynomial-time algorithm for the unweighted case [268, 315]. Note that for a different definition of the minimum spanning tree problem in a hypergraph, Tomescu and Zimand [369] have shown that the problem is NP-hard for hypergraphs with edges of size 3.

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Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria, Australia

    Marcus Brazil

  2. Department of Computer Science (DIKU), University of Copenhagen, Copenhagen, Denmark

    Martin Zachariasen

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Brazil, M., Zachariasen, M. (2015). Steiner Trees in Graphs and Hypergraphs. In: Optimal Interconnection Trees in the Plane. Algorithms and Combinatorics, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-319-13915-9_5

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