Query Complexity of the Metric Steiner Tree Problem

TL;DR

The first result shows that any (randomized) algorithm that estimates the Steiner Tree cost to within a $(5/3 - \varepsilon)$-factor requires $\Omega(n^2)$ queries, even if $k$ is a constant, and complements this result by showing an $\tilde{\Omega}(n + k^{6/5})$ query lower bound.

Abstract

We study the query complexity of the metric Steiner Tree problem, where we are given an $n \times n$ metric on a set $V$ of vertices along with a set $T \subseteq V$ of $k$ terminals, and the goal is to find a tree of minimum cost that contains all terminals in $T$. The query complexity for the related minimum spanning tree (MST) problem is well-understood: for any fixed $\varepsilon > 0$, one can estimate the MST cost to within a $(1+\varepsilon)$-factor using only $\tilde{O}(n)$ queries, and this is known to be tight. This implies that a $(2 + \varepsilon)$-approximate estimate of Steiner Tree cost can be obtained with $\tilde{O}(k)$ queries by simply applying the MST cost estimation algorithm on the metric induced by the terminals. Our first result shows that any (randomized) algorithm that estimates the Steiner Tree cost to within a $(5/3 - \varepsilon)$-factor requires $Ω(n^2)$ queries, even if $k$ is a constant. This lower bound is in sharp contrast to an upper bound of $O(nk)$ queries for computing a $(5/3)$-approximate Steiner Tree, which follows from previous work by Du and Zelikovsky. Our second main result, and the main technical contribution of this work, is a sublinear query algorithm for estimating the Steiner Tree cost to within a strictly better-than-$2$ factor, with query complexity $\tilde{O}(n^{12/7} + n^{6/7}\cdot k)=\tilde{O}(n^{13/7})=o(n^2)$. We complement this result by showing an $\tildeΩ(n + k^{6/5})$ query lower bound for any algorithm that estimates Steiner Tree cost to a strictly better than $2$ factor. Thus $\tildeΩ(n^{6/5})$ queries are needed to just beat $2$-approximation when $k = Ω(n)$; a sharp contrast to MST cost estimation where a $(1+o(1))$-approximate estimate of cost is achievable with only $\tilde{O}(n)$ queries.

Figures (9)

• Figure 1: An illustration of the trade-off between query complexity and approximation ratio for the metric Steiner Tree problem. The red curve shows the complexity of computing a Steiner Tree, while the green curve shows the complexity of estimating the minimum metric Steiner Tree cost. The upper bound at $(5/3)$-approximation follows from du1995component and zelikovsky199311; the bottom green curve (showing $\Theta(k)$ query complexity for $\alpha$-estimating the cost where $\alpha>2$) is due to czumaj2009estimating, and all other curves are results of this paper. All terms are inside a $\tilde{O}(\cdot)$ symbol.
• Figure 2: An illustration of edge replacement in Case 1.
• Figure 3: A schematic view of vertices and edges in Case 2.
• Figure 4: An illustration of edge replacement in Case 2.
• Figure 5: An illustration of edge replacement in Case 3.1 (assume that $u^*=u'_1$).

Theorems & Definitions (58)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 2.1: Chernoff Bound
• Claim 3.1
• Claim 3.2
• proof
• Claim 3.3