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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
If the input points alone are used as endpoints of the network edges, then the shortest network is their minimum spanning tree. However, shorter networks can often be obtained by adding Steiner points, and using both the new points and the input points as edge endpoints. [1] Another problem that uses Steiner points is Steiner triangulation. The ...
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.
The Steiner ratio is the supremum, over all point sets, of the ratio of lengths of the Euclidean minimum spanning tree to the Steiner minimum tree. Because the Steiner minimum tree is shorter, this ratio is always greater than one. [2] A lower bound on the Steiner ratio is provided by three points at the vertices of an equilateral triangle of ...
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.