In the classical Steiner tree problem, given an undirected, connected graph G=(V,E) with nonnegative edge costs and a set of terminals T⊆V, the objective is to find a minimumcost tree E′⊆E that spans the terminals. The problem is APXhard; the best known approximation algorithm has a ratio of ρ=ln(4)+ε<1.39. In this paper, we study a natural generalization, the multilevel Steiner tree (MLST) problem: given a nested sequence of terminals Tℓ⊂⋯⊂T1⊆V, compute nested trees Eℓ⊆⋯⊆E1⊆E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names including Multilevel Network Design, QualityofService Multicast tree, GradeofService Steiner tree, and MultiTier tree. Several approximation results are known. We first present two simple O(ℓ)approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most 2ℓ Steiner tree computations. We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present various integer linear programming (ILP) formulations for the MLST problem, and compare their running times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to ℓ=100 levels, which is sufficient for most applications such as network visualization or designing multilevel infrastructure.
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Multilevel Steiner Trees
In the classical Steiner tree problem, given an undirected, connected graph G =( V , E ) with nonnegative edge costs and a set of terminals T ⊆ V , the objective is to find a minimumcost tree E &prime ⊆ E that spans the terminals. The problem is APXhard; the bestknown approximation algorithm has a ratio of ρ = ln (4)+ε < 1.39. In this article, we study a natural generalization, the multilevel Steiner tree (MLST) problem: Given a nested sequence of terminals T ℓ ⊂ … ⊂ T 1 ⊆ V , compute nested trees E ℓ ⊆ … ⊆ E 1 ⊆ E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names, including Multilevel Network Design, QualityofService Multicast tree, GradeofService Steiner tree, and Multitier tree. Several approximation results are known. We first present two simple O (ℓ)approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most 2ℓ Steiner tree computations. We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present several integer linear programming formulations for the MLST problem and compare their running times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to ℓ = 100 levels, which is sufficient for most applications, such as network visualization or designing multilevel infrastructure.
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 Award ID(s):
 1740858
 NSFPAR ID:
 10294605
 Date Published:
 Journal Name:
 ACM Journal of Experimental Algorithmics
 Volume:
 24
 ISSN:
 10846654
 Page Range / eLocation ID:
 1 to 22
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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