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We present a simple and efficient O(m)-time 2-approximation algorithm for the maximum-leaf spanning tree problem, which is NP-complete even for planar graphs with maximum degree 4.

Abstract

The paper presents a simple and efficient 2-approximation algorithm for the maximum-leaf spanning tree problem. The key highlights are:
The algorithm has a single simple loop that can be implemented to run in O(m) time using basic data structures like arrays and linked lists.
The proof for the 2-approximation ratio is shorter and simpler compared to the previous best known 2-approximation algorithm.
The algorithm follows a greedy approach, expanding the tree in each round to maximize the number of leaves. If a round cannot increase the number of leaves, it looks ahead one round and attempts to maximize the number of leaves in two consecutive rounds.
When no suitable expansion vertices can be found for two consecutive rounds, the algorithm employs a "depth-first" expansion strategy to simplify the analysis of the approximation ratio.
The analysis shows that the approximation ratio is tight by providing an example where the algorithm returns a spanning tree with half the number of leaves as the optimal spanning tree.

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Key Insights Distilled From

by I-Cheng Liao... at **arxiv.org** 04-02-2024

Deeper Inquiries

The maximum-leaf spanning tree problem has various potential applications in real-world scenarios. One application is in communication networks, where maximizing the number of leaves in a spanning tree can help optimize the efficiency of data transmission and network connectivity. In circuit layouts, this problem can be utilized to design circuits with minimal delay and optimal routing paths. Additionally, in computer graphics, maximizing the number of leaves in a spanning tree can aid in creating efficient rendering algorithms and optimizing the display of graphical elements.

While the proposed algorithm provides a 2-approximation for the maximum-leaf spanning tree problem, there is potential to improve the approximation ratio by modifying the vertex selection strategy. One approach could be to prioritize vertices with specific characteristics, such as high degrees or unique structural properties, to enhance the algorithm's ability to maximize the number of leaves in the spanning tree. By refining the criteria for selecting expansion vertices, it may be possible to achieve a better approximation ratio than the current 2-approximation.

In comparison to other heuristic or exact approaches for the maximum-leaf spanning tree problem, the proposed algorithm stands out for its simplicity, efficiency, and 2-approximation guarantee. While exact algorithms may provide optimal solutions, they often come with higher computational complexity, making them less practical for large-scale graphs. On the other hand, heuristic approaches may sacrifice optimality for speed but may not guarantee a specific approximation ratio. The proposed algorithm strikes a balance by offering a reasonable approximation ratio of 2 while maintaining a linear-time complexity, making it a competitive choice for practical applications where a balance between accuracy and efficiency is crucial.

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