Steiner Trees in Industry

4 Conclusion The following problems are worth studying: Open problems on the Steiner ratio, such as Chung-Gilbert’s conjecture, Graham-Hwang’s conjecture, and Cielick’s conjecture, etc.. Find better approximation for network Steiner trees and establish an explicit lower bound for the approximation performance ratio of network Steiner trees. Close the gap between the lower bound and the upper bound for the approximation performance ratio of Steiner minimum arborescence. Find more efficient approximation algorithms for on-line and dynamic Steiner minimum trees and various Steiner tree packing problems. Find good approximation algorithms for multi-weighted Steiner trees and multiphase Steiner trees and study close relationship between multi-weighted Steiner trees, multiphase Steiner trees, and phylogenetic trees. Make clear whether there exists a polynomial-time approximation scheme for class Steiner tree in the special case with the real world background and highway interconnection problem. Close the gap between the lower bound and the upper bound for the approximation performance ratio of bottleneck Steiner tree in the Euclidean plane and make clear whether there exists a polynomial-time approximation scheme for the Steiner tree with minimum number of Steiner points and bounded edge-length. Implement efficient approximation algorithms to meet the requests from industries. We believe that to attack these new and old open problems new techniques are still required and the Steiner tree is still an attractive topic for researchers in combinatorial optimization and computer science.