# 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 T ⊆ V, the terminals. The objective is to find a minimum-cost edge set E' ⊆ E that spans the terminals. The problem is APX-hard [Bern & Plassman, IPL 1989]; the best known approximation algorithm has a ratio of ρ = (4)+ε < 1.39 [Byrka et al., J. ACM 2013]. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T_1 ⊂...⊂ T_k ⊆ V, compute nested edge sets E_1 ⊆...⊆ E_k ⊆ E that span the corresponding terminal sets with minimum total cost. (Note that, for ℓ=1,...,k, edges in E_ℓ contribute (k-ℓ+1)-fold to this cost). The MLST problem and variants thereof have been studied under names such as Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, Multi-Tier tree, etc. 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 Erdős--Rényi, random geometric, Watts--Strogatz, and Barabási--Albert network generation models for varying |V|, k, and terminal selection methods. 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 up to k < 22.

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