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A series of papers provided approximation algorithms for the minimum Steiner tree problem with approximation ratios that improved upon the 2 − 2/t ratio. This sequence culminated with Robins and Zelikovsky's algorithm in 2000 which improved the ratio to 1.55 by iteratively improving upon the minimum cost terminal spanning tree.
The RSMT is an NP-hard problem, and as with other NP-hard problems, common approaches to tackle it are approximate algorithms, heuristic algorithms, and separation of efficiently solvable special cases. An overview of the approaches to the problem may be found in the 1992 book by Hwang, Richards and Winter, The Steiner Tree Problem. [3]
Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. [2] (The minimum spanning tree for an entire graph is solvable in polynomial time.) Modularity maximization [5] Monochromatic triangle [3]: GT6 Pathwidth, [6] or, equivalently, interval thickness, and vertex separation number [7] Rank coloring; k-Chinese postman
Example of Steiner points (in red) added to a triangulation to improve the quality of triangles. In computational geometry , a Steiner point is a point that is not part of the input to a geometric optimization problem but is added during the solution of the problem, to create a better solution than would be possible from the original points alone.
The optimal solutions to the Steiner tree problem and the minimum Wiener connector can differ. Define the set of query vertices Q by Q = {v 1, ..., v 10}.The unique optimal solution to the Steiner tree problem is Q itself, which has Wiener index 165, whereas the optimal solution for the minimum Wiener connector problem is Q ∪ {r 1, r 2}, which has Wiener index 142.
Ding-Zhu Du (born May 21, 1948) is a Professor in the Department of Computer Science at The University of Texas at Dallas. [1] He is known for his research on the Euclidean minimum Steiner trees, [2] including an attempted proof of Gilbert–Pollak conjecture on the Steiner ratio, and the existence of a polynomial-time heuristic with a performance ratio bigger than the Steiner ratio.
The main motivation for studying the Hanan grid stems from the fact that it is known to contain a minimum length rectilinear Steiner tree for S. [1] It is named after Maurice Hanan, who was first [ 2 ] to investigate the rectilinear Steiner minimum tree and introduced this graph.
The goal of the Steiner tree problem is to connect these terminals by a tree whose weight is as small as possible. To transform this problem into an instance of the k-minimum spanning tree problem, Ravi et al. (1996) attach to each terminal a tree of zero-weight edges with a large number t of vertices per tree.