Sublinear Metric Steiner Forest via Maximal Independent Set

TL;DR

The paper addresses sublinear-time algorithms for the Metric Steiner Forest problem in a distance-query setting, introducing a framework that achieves an $O(\log k)$-approximation with $\widetilde{O}(n^{3/2})$ queries. A key methodological advance is a sublinear MIS (maximal independent set) estimator in the adjacency matrix model, which provides a purely multiplicative $(1+\varepsilon)$-approximation in $\widetilde{O}(n^{3/2}/\varepsilon^2)$ time and extends MIS techniques beyond bounded-degree graphs. The main algorithm builds a hierarchical scheme over threshold graphs and active-ball constructs, reducing Steiner Forest estimation to assembling within clusters induced by MIS centers; this yields the overall MIS-based $O(\log k)$-approximation for Steiner Forest. The paper also proves the $O(\log k)$ factor is tight for this MIS-based approach via two crafted instances, and it provides a detailed sublinear MIS machinery including single-sample and multi-sample estimators grounded in probabilistic ballot-theorem arguments. Together, these results broaden the sublinear algorithm toolbox to include Steiner-type network design in metric spaces and connect MIS estimation to sublinear Steiner-type problems.

Abstract

In this work we consider the Metric Steiner Forest problem in the sublinear time model. Given a set $V$ of $n$ points in a metric space where distances are provided by means of query access to an $n\times n$ distance matrix, along with a set of $k$ terminal pairs $(s_1,t_1), \dots, (s_k,t_k)\in V\times V$, the goal is to find a minimum-weight subset of edges that connects each terminal pair. Although sublinear time algorithms have been studied for estimating the weight of a minimum spanning tree in both general and metric settings, as well as for the metric Steiner Tree problem, no sublinear time algorithm was known for the metric Steiner Forest problem. Here, we give an $O(\log k)$-approximation algorithm for the problem that runs in time $\widetilde{O}(n^{3/2})$. Along the way, we provide the first sublinear-time algorithm for estimating the size of a Maximal Independent Set (MIS). Our algorithm runs in time $\widetilde{O}(n^{3/2}/\varepsilon^2)$ under the adjacency matrix oracle model and obtains a purely multiplicative $(1+\varepsilon)$-approximation. Previously, sublinear-time algorithms for MIS were only known for bounded-degree graphs.

Figures (2)

• Figure 1: An example of $I_1$ with $15$ vertices. The endpoints of line segments denote terminal pairs.
• Figure 2: An example of $I_2$ with $15$ vertices. The endpoints of line segments denote terminal pairs.

Theorems & Definitions (30)

• Definition 1.1: Sublinear Metric Steiner Forest
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof : Proof of \ref{['thm:steiner-forest-main']}
• Lemma 2.3: Connectivity within clusters
• proof
• Lemma 2.4: Connecting target pairs
• proof