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Structural Properties of Minimum Multi-source Multi-Sink Steiner Networks in the Euclidean Plane

Author

Listed:
  • Alexander Westcott

    (University of Melbourne)

  • Marcus Brazil

    (University of Melbourne)

  • Charl Ras

    (University of Melbourne)

Abstract

Given two finite sets A and B of points in the Euclidean plane, a minimum multi-source multi-sink Steiner network in the plane, or a minimum (A, B)-network, is a directed graph embedded in the plane with a dipath from every node in A to every node in B such that the total length of all arcs in the network is minimised. Such a network may contain Steiner points—nodes appearing in the solution that are neither in A nor B. We show that for any finite point sets A, B in the plane, there exists a minimum (A, B)-network that is constructible by straightedge and compass (this was claimed in a paper by Maxwell and Swanepoel, but their argument is incorrect). We use this property to formulate an algorithmic framework for exactly finding minimum (A, B)-networks in the Euclidean plane. We also present several new structural and geometric properties of minimum (A, B)-networks. In particular, we resolve a conjecture of Alfaro by proving that, for any terminal set A, adding an appropriate orientation to the edges of a minimum 2-edge-connected Steiner network on A yields a minimum (A, A)-network.

Suggested Citation

  • Alexander Westcott & Marcus Brazil & Charl Ras, 2023. "Structural Properties of Minimum Multi-source Multi-Sink Steiner Networks in the Euclidean Plane," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1104-1139, June.
    • Handle: RePEc:spr:joptap:v:197:y:2023:i:3:d:10.1007_s10957-023-02203-6
    DOI: 10.1007/s10957-023-02203-6
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    References listed on IDEAS

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    1. M. Brazil & D.A. Thomas & J.F. Weng, 2000. "On the Complexity of the Steiner Problem," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 187-195, June.
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