Approximations for two variants of the Steiner tree problem in the Euclidean plane

Given n terminals in the Euclidean plane and a positive constant l , find a Steiner tree T interconnecting all terminals with the minimum total cost of Steiner points and a specific material used to construct all edges in T such that the Euclidean length of each edge in T is no more than l . In this...

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Vydáno v:Journal of global optimization Ročník 57; číslo 3; s. 783 - 801
Hlavní autoři: Li, Jianping, Wang, Haiyan, Huang, Binchao, Lichen, Junran
Médium: Journal Article
Jazyk:angličtina
Vydáno: Boston Springer US 01.11.2013
Springer
Témata:
Algorithms
Computer Science
Fiber optic networks
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Real Functions
Very-large-scale integration
Specific material
Approximation algorithms
Subdivision
Packing
Steiner tree
ISSN:0925-5001, 1573-2916
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Shrnutí:Given n terminals in the Euclidean plane and a positive constant l , find a Steiner tree T interconnecting all terminals with the minimum total cost of Steiner points and a specific material used to construct all edges in T such that the Euclidean length of each edge in T is no more than l . In this paper, according to the cost b of each Steiner point and the different costs of some specific materials with the different lengths, we study two variants of the Steiner tree problem in the Euclidean plane as follows: (1) If a specific material to construct all edges in such a Steiner tree has its infinite length and the cost of per unit length of such a specific material used is c 1 , the objective is to minimize the total cost of the Steiner points and such a specific material used to construct all edges in T , i.e., , where T is a Steiner tree constructed, k 1 is the number of Steiner points and w ( e ) is the length of part cut from such a specific material to construct edge e in T , and we call this version as the minimum-cost Steiner points and edges problem (MCSPE, for short). (2) If a specific material to construct all edges in such a Steiner tree has its finite length L ( l  ≤ L ) and the cost of per piece of such a specific material used is c 2 , the objective is to minimize the total cost of the Steiner points and the pieces of such a specific material used to construct all edges in T , i.e., , where T is a Steiner tree constructed, k 2 is the number of Steiner points in T and k 3 is the number of pieces of such a specific material used, and we call this version as the minimum-cost Steiner points and pieces of specific material problem (MCSPPSM, for short). These two variants of the Steiner tree problem are NP -hard with some applications in VLSI design, WDM optical networks and wireless communications. In this paper, we first design an approximation algorithm with performance ratio 3 for the MCSPE problem, and then present two approximation algorithms with performance ratios 4 and 3.236 for the MCSPPSM problem, respectively.
ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-012-9967-3