Annals of Operations Research

, Volume 33, Issue 6, pp 481–499

A variational approach to the Steiner network problem

• J. H. Rubinstein
• D. A. Thomas
Section VI Steiner Tree Networks

Abstract

Supposen points are given in the plane. Their coordinates form a 2n-vectorX. To study the question of finding the shortest Steiner networkS connecting these points, we allowX to vary over a configuration space. In particular, the Steiner ratio conjecture is well suited to this approach and short proofs of the casesn=4, 5 are discussed. The variational approach was used by us to solve other cases of the ratio conjecture (n=6, see [11] and for arbitraryn points lying on a circle). Recently, Du and Hwang have given a beautiful complete solution of the ratio conjecture, also using a configuration space approach but with convexity as the major idea. We have also solved Graham's problem to decide when the Steiner network is the same as the minimal spanning tree, for points on a circle and on any convex polygon, again using the variational method.

Keywords

Variational Method Span Tree Variational Approach Minimal Span Tree Configuration Space
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. [1]
D.Z. Du, F.K. Hwang and E.N. Yao, The Steiner ratio conjecture is true for five points, J. Comb. Theory Ser. A 38(1985)230–240.Google Scholar
2. [2]
D.Z. Du, E.N. Yao and F.K. Hwang, A short proof of a result of Pollak on Steiner minimal trees, J. Comb. Theory Ser. A 32(1982)396–400.Google Scholar
3. [3]
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
4. [4]
E.N. Gilbert and H.O. Pollak, Steiner minimal trees, SIAM J. Appl. Math. 16(1968)1–29.Google Scholar
5. [5]
M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56(1981)179–195.Google Scholar
6. [6]
J.B. Kruskal, Jr., On the shortest spanning subtree of a graph and the travelling salesman problem, Proc. Amer. Math. Soc. 7(1956)48–50.Google Scholar
7. [7]
W. Meeks and S.T. Yau, Topology of three-dimensional manifolds and the embedding problem in minimal surface theory, Ann. Math. (2) 112(1980)441–485.Google Scholar
8. [8]
Z.A. Melzak, On the problem of Steiner, Can. Math. Bull. 4(1961)143–148.Google Scholar
9. [9]
H.O. Pollak, Some remarks on the Steiner problem, J. Comb. Theory Ser. A (1978)278–295.Google Scholar
10. [10]
R.C. Prim, Shortest connection networks and some generalizations, Bell. Syst. Tech. J. 36(1957)1389–1401.Google Scholar
11. [11]
J.H. Rubinstein and D.A. Thomas, The Steiner ratio conjecture for six points, J. Comb. Theory Ser. A, to appear.Google Scholar
12. [12]
J.H. Rubinstein and D.A. Thomas, Critical points for the Steiner ratio conjecture, Preprint.Google Scholar