
MultiLevel Steiner Trees
In the classical Steiner tree problem, one is given an undirected, conne...
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Kruskalbased approximation algorithm for the multilevel Steiner tree problem
We study the multilevel Steiner tree problem: a generalization of the S...
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A General Framework for Multilevel Subsetwise Graph Sparsifiers
Given an undirected weighted graph $G(V,E)$, a subsetwise sparsifier ove...
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A GridBased Approximation Algorithm for the Minimum Weight Triangulation Problem
Given a set of n points on a plane, in the Minimum Weight Triangulation ...
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Dense Steiner problems: Approximation algorithms and inapproximability
The Steiner Tree problem is a classical problem in combinatorial optimiz...
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Application of the Level2 Quantum Lasserre Hierarchy in Quantum Approximation Algorithms
The Lasserre Hierarchy is a set of semidefinite programs which yield inc...
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Approximation of trees by selfnested trees
The class of selfnested trees presents remarkable compression propertie...
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Computing VertexWeighted MultiLevel Steiner Trees
In the classical vertexweighted Steiner tree problem (VST), one is given an undirected graph G = (V,E) with nonnegative vertex weights, and a set T ⊆ V of terminals. The objective is to compute a minimumweight tree that spans T. The VST problem is NPhard and it is NPhard to approximate VST to within factor (1ε) T, but nearlybest approximation algorithms exist including the 2 Tapproximation algorithm of [Klein& Ravi, 1995]. Steiner tree problems and their variants have many applications, from combinatorial optimization and network routing to geometry and visualization. In some applications, the terminals may have different levels, priorities, or rateofservice requirements. For problems of this type, we study a natural generalization of the VST problem to multiple levels, referred to as the vertexweighted, multilevel Steiner tree (VMLST) problem: given a vertexweighted graph G = (V,E) and ℓ> 2 nested terminal sets T_ℓ⊂ T_ℓ1⊂...⊂ T_1⊆ V, compute a nested set of trees G_ℓ⊆ G_ℓ1⊆...⊆ G_1 where each tree G_i spans its corresponding terminal set T_i, with minimum total weight. We introduce a simple heuristic with approximation ratio O(ℓ T_1), which runs in a topdown manner using a singlelevel VST subroutine. We then show that the VMLST problem can be approximated to within 2 T_1 of the optimum with a greedy algorithm that connects "levelrespecting trees" at each iteration with a minimum costtoconnectivity ratio. This result is counterintuitive as it suggests that the seemingly harder multilevel version is not indeed harder than the singlelevel VST problem to approximate. The key tool in the analysis of our greedy approximation algorithm is a new "tailed spider decomposition."
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