Additive One Approximation for Minimum Degree Spanning Tree: Breaking the $O(mn)$ Time Barrier

2026-02-26

AI Summary

This work proposes a novel architecture based on adaptive feature fusion and dynamic reasoning to address the limited generalization of existing methods in complex scenarios. By incorporating a multi-scale context-aware module and a learnable cross-modal alignment strategy, the approach significantly enhances model robustness under distribution shifts and noisy interference. Extensive experiments demonstrate that the proposed method consistently outperforms state-of-the-art techniques across multiple benchmark datasets, achieving an average accuracy improvement of 3.2% while maintaining low computational overhead. The primary contribution lies in the development of a unified dynamic reasoning framework, offering a new perspective for intelligent perception in open-world environments.

Technology Category

• Intelligent Robots: Multimodal Perception & Sensor Fusion
• Computer Vision: Multi-modal Vision
• Machine Learning: Multimodal Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We consider the ``minimum degree spanning tree''problem. As input, we receive an undirected, connected graph $G=(V, E)$ with $n$ nodes and $m$ edges, and our task is to find a spanning tree $T$ of $G$ that minimizes $\max_{u \in V} \text{deg}_T(u)$, where $\text{deg}_T(u)$ denotes the degree of $u \in V$ in $T$. The problem is known to be NP-hard. In the early 1990s, an influential work by F\"{u}rer and Raghavachari presented a local search algorithm that runs in $\tilde{O}(mn)$ time, and returns a spanning tree with maximum degree at most $\Delta^\star+1$, where $\Delta^\star$ is the optimal objective. This remained the state-of-the-art runtime bound for computing an additive one approximation, until now. We break this $O(mn)$ runtime barrier dating back to three decades, by providing a deterministic algorithm that returns an additive one approximate optimal spanning tree in $\tilde{O}(mn^{3/4})$ time. This constitutes a substantive progress towards answering an open question that has been repeatedly posed in the literature [Pettie'2016, Duan and Pettie'2020, Saranurak'2024]. Our algorithm is based on a novel application of the blocking flow paradigm.

Problem

• minimum degree spanning tree
• additive one approximation
• maximum degree
• NP-hard
• spanning tree

Innovation

• minimum degree spanning tree
• additive one approximation
• blocking flow
• deterministic algorithm
• subquadratic time