Multi-Level Steiner Trees

In the classical Steiner tree problem, given an undirected, connected graph G=(V,E) with non-negative edge costs and a set of terminals Tsubseteq V, the objective is to find a minimum-cost tree E' subseteq E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of ρ = ln(4)+varepsilon < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals Tℓ subset dots subset T1 subseteq V, compute nested trees Eℓsubseteq dots subseteq E1subseteq E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier 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 multi-level infrastructure.