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The role of Steiner hulls in the solution to Steiner tree problems

  • Section VII Computational Geometry And TND
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Abstract

ASteiner tree problem on the plane is that of finding a minimum lengthSteiner tree connecting a given setK ofterminals and lying within a given regionR of the Euclidean plane; it includes as special cases the Euclidean Steiner minimal tree problem (ESMT), the rectilinear Steiner tree problem (RST), and the Steiner tree problem on graphs (STG). ASteiner hull forK inR generically refers to any subregion ofR known to contain a Steiner tree. This paper gives a survey of the role of Steiner hulls in the Steiner tree problem. The significance of Steiner hulls in the efficient solution of Steiner tree problems is outlined, and then a compendium is given of the known Steiner hull constructions for ESMT, RST, and STG problems.

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References

  1. A.V. Aho, M.R. Garey and H.W. Hwang, Rectilinear Steiner trees: Efficient special-case algorithms, Networks 7(1977)37–58.

    Google Scholar 

  2. A. Balakrishnan and W.R. Patel, Problem reduction methods and a tree generation algorithm for the Steiner tree problem, Networks 17(1987)65–85.

    Google Scholar 

  3. J.E. Beasley, An algorithm for the Steiner problem in graphs, Networks 14(1984)147–159.

    Google Scholar 

  4. M. Bern, Faster exact algorithms for Steiner trees in planar networks, Networks 20(1990)109–120.

    Google Scholar 

  5. M. Bern and D. Bienstock, Polynomially solvable special cases of the Steiner problem in planar networks, Ann. Oper. Res., this volume.

  6. M. Bern, E.L. Lawler and R.T. Wong, Linear time computation of optimal subgraphs of decomposable graphs, J. Alg. 8(1987)216–235.

    Google Scholar 

  7. D. Bienstock and C.L. Monma, On the complexity of covering the faces of a planar graph, SIAM J. Comput. 17(1988)53–76.

    Google Scholar 

  8. D. Bienstock and C.L. Monma, Optimal enclosing regions in planar graphs, Networks 19(1989)79–94.

    Google Scholar 

  9. F.R.K. Chung and R.L. Graham, Steiner tree for ladders, Ann. Discr. Math. 2(1978)173–200.

    Google Scholar 

  10. E.J. Cockayne, On the efficiency of the algorithm for Steiner minimal trees, SIAM J. Appl. Math. 18(1970)150–159.

    Google Scholar 

  11. E.W. Dreyfus and R.A. Wagner, The Steiner problem in graphs, Networks 1(1972)195–207.

    Google Scholar 

  12. C.W. Duin and A. Volgenant, An edge elimination test for the Steiner problem in graphs, Oper. Res. Lett. 8(1989)79–83.

    Google Scholar 

  13. C.W. Duin and A. Volgenant, Reduction tests for the Steiner problem in graphs, Networks 19(1989)549–567.

    Google Scholar 

  14. R.E. Erickson, C.L. Monma and A.F. Veinott, Send and split method for minimum-concave-cost network flows, Math. Oper. Res. 12(1987)634–664.

    Google Scholar 

  15. M.R. Garey, R.L. Graham and D.S. Johnson, The complexity of computing Steiner minimal trees, SIAM J. Appl. Math. 32(1977)835–859.

    Google Scholar 

  16. M.R. Garey and D.S. Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math. 32(1977)826–834.

    Google Scholar 

  17. E.N. Gilbert and H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16(1968)1–29.

    Google Scholar 

  18. R.L. Graham, An efficient algorithm for determining the convex hull of a set of points, Info. Proc. Lett. 1(1972)132–133.

    Google Scholar 

  19. S.L. Hakimi, Steiner's problem in graphs and its implications, Networks 1(1971)113–133.

    Google Scholar 

  20. M. Hanan, On Steiner's problem with rectilinear distance, SIAM J. Appl. Math. 14(1966)255–265.

    Google Scholar 

  21. F.K. Hwang, G.D. Song, G.Y. Ting and D.Z. Hu, A decomposition theorem on Euclidean Steiner minimal trees, Discr. Comp. Geom. 3(1988)367–382.

    Google Scholar 

  22. L. Kou, G. Markowski and L. Berman, A fast algorithm for Steiner trees, Acta Informatica 15(1981)141–145.

    Google Scholar 

  23. A.L. Liestman, Construction of Steiner trees with obstacles in the plane, Master Thesis, University of Illinois, Urbana-Champaign, Urbana, IL (1978).

    Google Scholar 

  24. Z.A. Melzak, On the problem of Steiner, Can. Math. Bull. 4(1961)143–148.

    Google Scholar 

  25. J.S. Provan, Convexity and the Steiner tree problem, Networks 18(1988)55–72.

    Google Scholar 

  26. J.S. Provan, An approximation scheme for finding Steiner trees with obstacles, SIAM J. Comput. 17(1988)920–934.

    Google Scholar 

  27. J.S. Provan, Finding shortest enclosing walks and cycles in embedded graphs, Info. Proc. Lett. 30(1989)119–125.

    Google Scholar 

  28. J.S. Provan, Two new Steiner hull techniques for the Steiner tree problem, Technical Report No. UNC/OR/TR-89/6, Operations Research Department, University of North Carolina at Chapel Hill, NC (1989), Algorithmica (1991), to appear.

  29. A. Rosenthal, Computing the reliability of complex networks, SIAM J. Appl. Math. 32(1977)384–393.

    Google Scholar 

  30. P.W. Shor and W.D. Smith, Steiner hulls andσ hulls, Manuscript (1982).

  31. J.M. Smith, Steiner minimal trees with obstacles, Technical Report, Department of Industrial Engineering and Operations Research, University of Massachusetts, Amherst, MA (1982).

    Google Scholar 

  32. J.M. Smith and P. Winter, Computational geometry and topological network design,Proc. NATO Advanced Research Workshop on Topological Network Design: Analysis and Synthesis, Gentofte, Denmark (1989).

  33. H. Takahashi and A. Matsuyama, An approximate solution for the Steiner problem in graphs, Math. Japonica 24(1980)573–577.

    Google Scholar 

  34. K. Takamizawa, T. Nishizeki and N. Saito, Linear-time computability of combinatorial problems on series-parallel graphs. J. ACM 29(1982)623–641.

    Google Scholar 

  35. J.A. Wald and C.J. Colbourn, Steiner trees, partial 2-trees, and minimum IFI networks, Networks 13(1983)159–167.

    Google Scholar 

  36. P. Winter, An algorithm for the Steiner problem in the Euclidean plane, Networks 15(1985)323–345.

    Google Scholar 

  37. P. Winter, Steiner problem in networks: A survey, Networks 17(1987)129–167.

    Google Scholar 

  38. Y.Y. Yang and O. Wing, Optimal and suboptimal algorithms for the wiring problem,Proc. IEEE Int. Symp. on Circuit Theory (1972), pp. 154–158.

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  1. Department of Operations Research, University of North Carolina, 27599-3180, Chapel Hill, North Carolina, USA

    J. Scott Provan

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Provan, J.S. The role of Steiner hulls in the solution to Steiner tree problems. Ann Oper Res 33, 537–548 (1991). https://doi.org/10.1007/BF02067240

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