Multi-Level Steiner Trees

In the classical Steiner tree problem, one is given an undirected, connected graph G=(V,E) with non-negative edge costs and a set of terminals T subseteq V. The objective is to find a minimum-cost edge set E' subseteq E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of rho = ln(4)+epsilon < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T_1 subset ... subset T_k subseteq V, compute nested edge sets E_1 subseteq ... subseteq E_k subseteq E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under names such as Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two natural heuristics with approximation factor O(k). Based on these, we introduce a composite algorithm that requires 2^k Steiner tree computations. We determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio and needs at most 2k Steiner tree computations. We compare five algorithms experimentally on several classes of graphs using four types of graph generators. We also implemented an integer linear program for MLST to provide ground truth. Our combined algorithm outperforms the others both in theory and in practice when the number of levels is small (k <= 22), which works well for applications such as designing multi-level infrastructure or network visualization.