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On the Full and Bottleneck Full Steiner Tree Problems
 In: Proceedings of the 9th Annual International Conference (COCOON 2003), Big Sky
, 2003
"... Abstract. Given a graph G = (V, E) with a length function on edges and a subset R of V, the full Steiner tree is defined to be a Steiner tree in G with all the vertices of R as its leaves. Then the full Steiner tree problem is to find a full Steiner tree in G with minimum length, and the bottleneck ..."
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Abstract. Given a graph G = (V, E) with a length function on edges and a subset R of V, the full Steiner tree is defined to be a Steiner tree in G with all the vertices of R as its leaves. Then the full Steiner tree problem is to find a full Steiner tree in G with minimum length, and the bottleneck full Steiner tree problem is to find a full Steiner tree T in G such that the length of the largest edge in T is minimized. In this paper, we present a new approximation algorithm with performance ratio 2ρ for the full Steiner tree problem, where ρ is the bestknown performance ratio for the Steiner tree problem. Moreover, we give an exact algorithm of O(E  log E) time to solve the bottleneck full Steiner tree problem. 1
A Better ConstantFactor Approximation for SelectedInternal Steiner Minimum Tree
 ALGORITHMICA
, 2009
"... The selectedinternal Steiner minimum tree problem is a generalization of original Steiner minimum tree problem. Given a weighted complete graph G = (V, E) with weight function c, and two subsets R ′ � R ⊆ V with R − R ′ ≥2, selectedinternal Steiner minimum tree problem is to find a minimum su ..."
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The selectedinternal Steiner minimum tree problem is a generalization of original Steiner minimum tree problem. Given a weighted complete graph G = (V, E) with weight function c, and two subsets R ′ � R ⊆ V with R − R ′ ≥2, selectedinternal Steiner minimum tree problem is to find a minimum subtree T of G interconnecting R such that any leaf of T does not belong to R ′. In this paper, suppose c is metric, we obtain a (1 + ρ)approximation algorithm for this problem, where ρ is the bestknown approximation ratio for the Steiner minimum tree problem.
A Polylogarithmic Approximation for Computing NonMetric Terminal Steiner Trees
"... The main contribution of this short note is to provide improved bounds on the approximability of constructing terminal Steiner trees in arbitrary undirected graphs. Technically speaking, our results are obtained by relating this computational task to that of computing group Steiner trees. As a secon ..."
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The main contribution of this short note is to provide improved bounds on the approximability of constructing terminal Steiner trees in arbitrary undirected graphs. Technically speaking, our results are obtained by relating this computational task to that of computing group Steiner trees. As a secondary objective, we make a concentrated effort to distinguish between the factor by which constructed trees exceed the optimal backbone cost and between the deviation from the optimal terminal linking cost.
Approximating Full Steiner Tree in a Unit Disk Graph
, 2014
"... Given an edgeweighted graph G = (V,E) and a subset R of V, a Steiner tree of G is a tree which spans all the vertices in R. A full Steiner tree is a Steiner tree which has all the vertices of R as its leaves. The full Steiner tree problem is to find a full Steiner tree of G with minimum weight. In ..."
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Given an edgeweighted graph G = (V,E) and a subset R of V, a Steiner tree of G is a tree which spans all the vertices in R. A full Steiner tree is a Steiner tree which has all the vertices of R as its leaves. The full Steiner tree problem is to find a full Steiner tree of G with minimum weight. In this paper we present a 20approximation algorithm for the full Steiner tree problem when G is a unit disk graph. 1
A Note on Online Steiner Tree Problems
"... We introduce and study a new Steiner tree problem variation called the bursty Steiner tree problem where new nodes arrive into bursts. This is an online problem which becomes the wellknown online Steiner tree problem if the number of nodes in each burst is exactly one and becomes the classical Ste ..."
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We introduce and study a new Steiner tree problem variation called the bursty Steiner tree problem where new nodes arrive into bursts. This is an online problem which becomes the wellknown online Steiner tree problem if the number of nodes in each burst is exactly one and becomes the classical Steiner tree problem if all the nodes that need to be connected appear in a single burst. In undirected graphs, we provide a tight bound of Θ(min{log k,m}) on the competitive ratio for this problem, where k is the total number of nodes to be connected and m is the total number of different bursts. In directed graphs of bounded edge asymmetry α, we provide a near tight competitive ratio for this problem. We also consider a bursty variation of the terminal Steiner tree problem and provide the upper bound of min{4ρ, 3λm} and the lower bound of min{ρ/2,m/4} on the competitive ratio in undirected complete graphs, where λ is the current best approximation for the terminal Steiner tree problem and ρ = 12 log k. These are the first such results which provide clear performance tradeoffs for the novel Steiner tree problem variations that subsume both of their online and classical versions. 1
Hardness and Approximation of the SelectedLeafTerminal Steiner Tree Problem
"... Abstract. For a complete graph G = (V, E) with length function l: E → R + and two vertex subsets R ⊂ V and R ′ ⊆ R, a selectedleafterminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R \ R ′ belong to the leaves of this Steiner tree. The selectedleaf ..."
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Abstract. For a complete graph G = (V, E) with length function l: E → R + and two vertex subsets R ⊂ V and R ′ ⊆ R, a selectedleafterminal Steiner tree is a Steiner tree which contains all vertices in R such that all vertices in R \ R ′ belong to the leaves of this Steiner tree. The selectedleafterminal Steiner tree problem is to find a selectedleafterminal Steiner tree T whose total lengths � (u,v)∈T l(u, v) is minimum. In this paper, we show that the problem is both NPcomplete and MAX SNPhard when the lengths of edges are restricted to either 1 or 2. We also provide an approximation algorithm for the problem. Keywords: Approximation algorithms, MAX SNPhard, Steiner tree, the selectedleafterminal Steiner tree problem, design and analysis
On the Hardness of Full Steiner Tree Problems
, 2014
"... Given a weighted graph G = (V,E) and a subset R of V, a Steiner tree in G is a tree which spans all vertices in R. The vertices in V \R are called Steiner vertices. A full Steiner tree is a Steiner tree in which each vertex of R is a leaf. The full Steiner tree problem is to find a full Steiner tree ..."
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Given a weighted graph G = (V,E) and a subset R of V, a Steiner tree in G is a tree which spans all vertices in R. The vertices in V \R are called Steiner vertices. A full Steiner tree is a Steiner tree in which each vertex of R is a leaf. The full Steiner tree problem is to find a full Steiner tree with minimum weight. The bottleneck full Steiner tree problem is to find a full Steiner tree which minimizes the length of the longest edge. The kbottleneck full Steiner tree problem is to find a bottleneck full Steiner tree with at most k Steiner vertices. The smallest full Steiner tree problem is to find a full Steiner tree with the minimum number of Steiner vertices. We present some results on the hardness of these problems. The presented reductions show the connection between the full Steiner tree, the group Steiner tree, and the connected set cover problems. 1