Better Learning-Augmented Spanning Tree Algorithms via Metric Forest Completion

2026-02-27

AI Summary

This work addresses the problem of efficiently constructing approximate minimum spanning trees (MSTs) in arbitrary metric spaces. Building upon the Metric Forest Completion (MFC) framework, the authors propose a learning-augmented approach that strategically selects representative points and restricts edge consideration to those incident to these points, thereby interpolating between optimal and fast approximation algorithms. Theoretical contributions include tightening the approximation factor of MFC from 2.62 to the information-theoretic lower bound of 2, and improving the approximation guarantee for degree-constrained MSTs from (2γ+1) to 2γ, while also enabling instance-dependent performance improvements. Empirical evaluations demonstrate the method’s superior practicality and efficiency, achieving subquadratic time complexity without sacrificing solution quality.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We present improved learning-augmented algorithms for finding an approximate minimum spanning tree (MST) for points in an arbitrary metric space. Our work follows a recent framework called metric forest completion (MFC), where the learned input is a forest that must be given additional edges to form a full spanning tree. Veldt et al. (2025) showed that optimally completing the forest takes $\Omega(n^2)$ time, but designed a 2.62-approximation for MFC with subquadratic complexity. The same method is a $(2\gamma + 1)$-approximation for the original MST problem, where $\gamma \geq 1$ is a quality parameter for the initial forest. We introduce a generalized method that interpolates between this prior algorithm and an optimal $\Omega(n^2)$-time MFC algorithm. Our approach considers only edges incident to a growing number of strategically chosen ``representative''points. One corollary of our analysis is to improve the approximation factor of the previous algorithm from 2.62 for MFC and $(2\gamma+1)$ for metric MST to 2 and $2\gamma$ respectively. We prove this is tight for worst-case instances, but we still obtain better instance-specific approximations using our generalized method. We complement our theoretical results with a thorough experimental evaluation.

Problem

• learning-augmented algorithms
• minimum spanning tree
• metric forest completion
• approximation algorithms
• metric space

Innovation

• learning-augmented algorithms
• metric forest completion
• minimum spanning tree
• approximation algorithms
• representative points