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Given a weighted graph G(V, E) and t ≥ 1, a subgraph H is a t–spanner of G if the lengths of shortest paths in G are preserved in H up to a multiplicative factor of t. The subsetwise spanner problem aims to preserve distances in G for only a subset of the vertices. We generalize the minimumcost subsetwise spanner problem to one where vertices appear on multiple levels, which we call the multilevel graph spanner (MLGS) problem, and describe two simple heuristics. Applications of this problem include road/network building and multilevel graph visualization, especially where vertices may require different grades of service. We formulate a 0–1 integer linear program (ILP) of size O(EV 2) for the more general minimum pairwise spanner problem, which resolves an open question by Sigurd and Zachariasen on whether this problem admits a useful polynomialsize ILP. We extend this ILP formulation to the MLGS problem, and evaluate the heuristic and ILP performance on random graphs of up to 100 vertices and 500 edges.

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 timesmore »