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 heuristicsmore »
Approximation algorithms for the vertexweighted gradeofservice Steiner tree problem
Given a graph G = (V, E) and a subset T ⊆ V of terminals, a Steiner tree of G is a tree that
spans T. In the vertexweighted Steiner tree (VST) problem, each vertex is assigned a nonnegative weight, and the goal is to compute a minimum weight Steiner tree of G. Vertexweighted problems have applications in network design and routing, where there are different costs for installing or maintaining facilities at different vertices. We study a natural generalization of the VST problem motivated by multilevel graph construction, the vertexweighted gradeofservice Steiner tree problem (VGSST), which can be stated as follows: given a graph G and terminals T, where each terminal v ∈ T requires a facility of a minimum grade of service R(v) ∈ {1, 2, . . . `}, compute a Steiner tree G0 by installing facilities on a
subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in G 0 with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multilevel variants such as this one can be useful in network design problems where vertices may require facilities more »
 Publication Date:
 NSFPAR ID:
 10109426
 Journal Name:
 ArXiv.org
 ISSN:
 23318422
 Sponsoring Org:
 National Science Foundation
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