Sublinear Metric Steiner Forest via Maximal Independent Set

2025-10-13

AI Summary

This paper studies the Steiner Forest problem in metric spaces under sublinear-time computation: given an $n$-point metric space (accessed only via queries to its $n imes n$ distance matrix) and $k$ terminal pairs, the goal is to construct a minimum-weight edge set connecting all pairs. We present the first sublinear-time approximation algorithm for general metric spaces—without assuming bounded doubling dimension or other geometric restrictions. Our approach operates in the adjacency-matrix query model and reduces Steiner Forest to estimating the maximum independent set (MIS) size. Crucially, we design the first sublinear-time $(1+varepsilon)$-approximation algorithm for MIS estimation in general graphs. This yields an $O(log k)$-approximation for Steiner Forest in $widetilde{O}(n^{3/2})$ time, and an $widetilde{O}(n^{3/2}/varepsilon^2)$-time $(1+varepsilon)$-approximation for MIS estimation. Our work overcomes prior limitations restricted to bounded-degree graphs and significantly broadens the applicability of sublinear algorithms to metric optimization problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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 $n imes n$ distance matrix, along with a set of $k$ terminal pairs $(s_1,t_1), dots, (s_k,t_k)in V imes V$, 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.

Problem

• Develops sublinear algorithm for Metric Steiner Forest problem
• Provides first sublinear approximation for Maximal Independent Set
• Addresses connectivity of terminal pairs in metric spaces

Innovation

• Sublinear time algorithm for metric Steiner Forest
• Maximal Independent Set estimation in sublinear time
• Approximation algorithm with logarithmic factor