Multi-Level Steiner Trees

TL;DR

A rudimentary composite algorithm is introduced that 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.

Abstract

In the classical Steiner tree problem, given an undirected, connected graph $G=(V,E)$ with non-negative edge costs and a set of \emph{terminals} $T\subseteq V$, 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 \emph{multi-level Steiner tree} (MLST) problem: given a nested sequence of terminals $T_{\ell} \subset \dots \subset T_1 \subseteq V$, compute nested trees $E_{\ell}\subseteq \dots \subseteq E_1\subseteq E$ 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 $O(\ell)$-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 $2\ell$ 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 $\ell=100$ levels, which is sufficient for most applications such as network visualization or designing multi-level infrastructure.

Figures (13)

• Figure 1: An illustration of an MLST with $\ell=3$ for the graph at the right. Solid and open circles represent terminal and non-terminal nodes, respectively. Note that the level-3 tree (left) is contained in the level-2 tree (mid), which is in turn contained in the level-1 tree (right).
• Figure 2: The approximation ratio of $\frac{\ell+1}{2}$ for the top-down approach is asymptotically tight. In the example above for $\ell=2$, the input graph (left) consists of a $(k+1)$-cycle with one edge of weight $k-\varepsilon$. The solution returned by top-down (right) has cost $\mathrm{TOP}\xspace = (k-\varepsilon) + (k-\varepsilon + k) \approx 3k$, whereas $\mathrm{OPT}\xspace = 2k$.
• Figure 3: The approximation ratio of $\ell$ for the bottom-up approach is asymptotically tight. Using the same input graph as in Figure \ref{['fig:examples-top']}, except by modifying the edge of weight $k-\varepsilon$ so that its weight is $1+\varepsilon$, we see that $\mathrm{BOT}\xspace = 2k$, whereas $\mathrm{OPT}\xspace = (1+\varepsilon) + (1+\varepsilon + k-1) \approx k+1$.
• Figure 4: Illustration of the composite heuristic for various subsets $\mathcal{Q}$, with $\ell=5$. Orange arrows pointing downward indicate propagation of edges similar to the top-down approach. Blue arrows pointing upward indicate pruning of unneeded edges, similar to the bottom-up approach.
• Figure 5: Approximation ratios for the composite algorithm for $\ell=1,\dots,100$ (black curve), compared to the ratio $t=e\rho$ (red dashed line) guaranteed by the algorithm of Charikar et al. 1288137 and $t=2.454\rho$ (green dashed line) guaranteed by the algorithm of Karpinski et al. Karpinski2005. The table to the right lists the exact values for the ratio $t/\rho$.

Theorems & Definitions (23)

• definition 1: Multi-Level Steiner Tree (MLST) Problem
• theorem 1
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof
• theorem 2
• proof