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Better Learning-Augmented Spanning Tree Algorithms via Metric Forest Completion

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Minimum spanning treemetric spaceslearning-augmented algorithmsalgorithms with predictionsapproximation algorithmsdynamic programmingk-center
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 Ω(n2)\Omega(n^2) time, but designed a 2.62-approximation for MFC with subquadratic complexity. The same method is a (2γ+1)(2\gamma + 1)-approximation for the original MST problem, where γ≥1\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 Ω(n2)\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 (2γ+1)(2\gamma+1) for metric MST to 2 and 2γ2\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.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
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Overview

Overall Novelty Assessment

The paper contributes a generalized algorithm for metric forest completion that interpolates between prior subquadratic and optimal quadratic-time methods, improving approximation factors from 2.62 to 2 for MFC and from (2γ+1) to 2γ for metric MST. It resides in the Metric Forest Completion Approaches leaf, which contains only one sibling paper (Approximate Forest Completion). This represents a sparse but focused research direction within the broader learning-augmented MST landscape, suggesting the forest completion paradigm is relatively nascent compared to other algorithmic frameworks in the field.

The taxonomy reveals that metric forest completion sits alongside query-based learning-augmented MST methods, which minimize edge queries under uncertainty rather than completing partial forests. The broader parent branch encompasses learning-augmented algorithms with predictions, distinguishing this work from MST-based regularization techniques (which use tree structures as geometric priors in representation learning) and domain-specific applications (depth enhancement, hyperbolic embeddings). The scope notes clarify that forest completion methods emphasize metric space properties and approximation guarantees, excluding query models and non-metric approaches that characterize neighboring research directions.

Among 14 candidates examined across three contributions, no refutable prior work was identified. The instance-specific approximation bounds contribution examined 4 candidates with none providing overlapping results, while the shared-budget multi-instance k-center algorithm examined 10 candidates, again with no refutations. The generalized multi-representative algorithm contribution had no candidates examined. This limited search scope—focused on top-K semantic matches—suggests the specific combination of representative-point selection and instance-aware bounds may be novel within the examined literature, though the small candidate pool precludes definitive claims about the broader field.

Based on the top-14 semantic matches and the sparse taxonomy leaf (two papers total), the work appears to advance a relatively young research direction. The absence of refutable candidates across examined contributions indicates potential novelty, but the limited search scale means unexplored prior work could exist outside the semantic neighborhood. The tight worst-case bounds and instance-specific analysis represent technical refinements that may distinguish this work from its single sibling paper, though comprehensive field coverage would require broader literature examination.

Taxonomy

Core-task Taxonomy Papers
7
3
Claimed Contributions
14
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: learning-augmented minimum spanning tree approximation in metric spaces. The field structure suggested by the taxonomy divides into three main branches. The first branch, Learning-Augmented MST Algorithms with Predictions, focuses on algorithmic frameworks that leverage machine-learned predictions or partial information to improve MST construction in metric settings, often addressing scenarios where exact distances are costly to query or only partially observable. The second branch, MST-Based Regularization and Representation Learning, explores how MST structures can serve as regularizers or geometric priors in learning tasks, guiding embeddings and feature extraction. The third branch, MST Applications in Specialized Domains, examines domain-specific uses of MST techniques, such as depth enhancement or hyperbolic embeddings, where the tree structure plays a supporting role in solving applied problems. Within the learning-augmented algorithms branch, a particularly active line of work centers on forest completion approaches, where the goal is to reconstruct or approximate a full MST given a partial forest or noisy edge information. Metric Forest Completion[0] and Approximate Forest Completion[5] both tackle this challenge, with the former emphasizing metric space guarantees and the latter exploring approximation trade-offs when exact completion is intractable. These works contrast with query-based methods like Query Policies MST[2], which focus on adaptively selecting which edges to reveal under budget constraints. Meanwhile, the regularization branch includes studies such as MST Regularization[6] and Graph Regularized Autoencoder[4], which use tree-based penalties to enforce geometric consistency in learned representations. Metric Forest Completion[0] sits squarely in the forest completion cluster, sharing methodological DNA with Approximate Forest Completion[5] but distinguished by its focus on provable metric approximations rather than heuristic or probabilistic guarantees.

Claimed Contributions

Generalized multi-representative MFC algorithm with improved approximation guarantees

The authors present a generalized approximation algorithm for metric forest completion that selects multiple representatives per component rather than just one. This method interpolates between the existing MFC-Approx algorithm and an optimal algorithm, achieving a 2-approximation for MFC and 2γ-approximation for metric MST, improving upon prior bounds of 2.62 and (2γ+1) respectively.

0 retrieved papers
Instance-specific approximation bounds via cost function

The authors introduce instance-specific approximation guarantees based on a cost function that measures the maximum distance between points and their nearest representatives. This provides tighter bounds than worst-case guarantees and can be efficiently computed in practice to serve as a proxy for true approximation quality.

4 retrieved papers
2-approximation algorithm for shared-budget multi-instance k-center problem

The authors formalize the Best Representatives problem as a generalization of k-center clustering where multiple instances share a budget for cluster centers. They develop a 2-approximation algorithm combining a classical greedy k-center approach with dynamic programming for budget allocation across instances.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Generalized multi-representative MFC algorithm with improved approximation guarantees

The authors present a generalized approximation algorithm for metric forest completion that selects multiple representatives per component rather than just one. This method interpolates between the existing MFC-Approx algorithm and an optimal algorithm, achieving a 2-approximation for MFC and 2γ-approximation for metric MST, improving upon prior bounds of 2.62 and (2γ+1) respectively.

Contribution

Instance-specific approximation bounds via cost function

The authors introduce instance-specific approximation guarantees based on a cost function that measures the maximum distance between points and their nearest representatives. This provides tighter bounds than worst-case guarantees and can be efficiently computed in practice to serve as a proxy for true approximation quality.

Contribution

2-approximation algorithm for shared-budget multi-instance k-center problem

The authors formalize the Best Representatives problem as a generalization of k-center clustering where multiple instances share a budget for cluster centers. They develop a 2-approximation algorithm combining a classical greedy k-center approach with dynamic programming for budget allocation across instances.

Table of Contents

Better Learning-Augmented Spanning Tree Algorithms via Metric Forest Completion | Novelty Validation