Annals of Operations Research

, Volume 238, Issue 1–2, pp 315–328

# An estimate of the objective function optimum for the network Steiner problem

• V. Kirzhner
• Z. Volkovich
• E. Ravve
• G.-W. Weber
Article

## Abstract

A complete weighted graph, $$G(X,\varGamma ,W)$$, is considered. Let $$\tilde{X}\subset X$$ be some subset of vertices and, by definition, a Steiner tree is any tree in the graph G such that the set of the tree vertices includes set $$\tilde{X}$$. The Steiner tree problem consists of constructing the minimum-length Steiner tree in graph G, for a given subset of vertices $$\tilde{X}$$ The effectively computable estimate of the minimal Steiner tree is obtained in terms of the mean value and the variance over the set of all Steiner trees. It is shown that in the space of the lengths of the graph edges, there exist some regions where the obtained estimate is better than the minimal spanning tree-based one.

## Keywords

Spanning tree Steiner tree Steiner problem

## References

1. Brazil, M., Thomas, D. A., & Weng, J. F. (2004). Upper and lower bounds for the lengths of Steiner trees in 3-space. Geometriae Dedicata, 109, 107–119.
2. Cieslik, Dietmar. (2003). The Steiner ratio of several discrete metric spaces. Discrete Mathematics, 260, 189–196.
3. Du, D.-Z., Lu, B., Ngo, H., & Pardalos, P. M. (2001). Steiner tree problems. In C. Floudas & P. Pardalos (Eds.), Encyclopedia of optimization (Vol. 5, pp. 227–290). Dordrecht: Kluwer Academic Publishers.Google Scholar
4. Garey, M. R., Graham, R. L., & Johnson, D. S. (1976). Some NP-complete geometric problems. In Eighth annual symposium on theory of computing, pp 10-22.Google Scholar
5. Gilbert, E. N., & Pollak, H. O. (1968). Steiner minimal trees. SIAM Journal on Applied Mathematics, 16, 11–29.Google Scholar
6. Hardy, G. H., Littlewood, J. E., & Polya, G. (1952). Inequalities (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
7. Hwang, F. K., Richards, D. S., & Winter, P. (1992). The Steiner tree problem. North-Holland: Elsevier.Google Scholar
8. Innami, N., Kim, B. H., Mashiko, Y., & Shiohama, K. (2010). The Steiner ratio conjecture of Gilbert-Pollak may still be open. Algorithmica, 57, 869–872.
9. Kirzhner, V., & Rublinecky, V. I. (1974). An upper limit for the traveling salesman minimal tour. In Proceeding low temperature physic institute “computing mathematics & Computers”, V 1974, pp 120–122 (in Russian).Google Scholar
10. Skiena, S. (1990). Implementing discrete mathematics: Combinatorics and graph theory with mathematica. Reading, MA: Addison-Wesley. p. 235.Google Scholar
11. Zaycev, I., Vayner, V., & Kirzhner, V. (1988). An estimate of the functional optimum in problems of connecting objects with a network. In Ukrainian Doklady, Series A (Vol. 8, pp. 71–74). (in Russian).Google Scholar

## Authors and Affiliations

• V. Kirzhner
• 1
• Z. Volkovich
• 2
• E. Ravve
• 2
• G.-W. Weber
• 3
• 4
• 5
• 6
• 7
1. 1.Institute of EvolutionUniversity of HaifaHaifaIsrael
2. 2.Ort Braude College of EngineeringKarmielIsrael
3. 3.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
4. 4.University of SiegenSiegenGermany
5. 5.University of AveiroAveiroPortugal