## Abstract

Given a graph \(G = (V, E)\), we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path problem. For the approximation version of this problem, a \(\tilde{O}(mn)\) time algorithm that computes a spanning tree of degree at most \(\varDelta ^*+1\) is previously known [Fürer & Raghavachari 1994]; here \(\varDelta ^*\) denotes the minimum tree degree of all the spanning trees. In this paper we give the first near-linear time approximation algorithm for this problem. Specifically speaking, we propose an \(\tilde{O}(\frac{1}{\epsilon ^7}m)\) time algorithm that computes a spanning tree with tree degree \((1+\epsilon )\varDelta ^*+ O(\frac{1}{\epsilon ^2}\log n)\) for any constant \(\epsilon \in (0,\frac{1}{6})\). Thus, when \(\varDelta ^*=\omega (\log n)\), we can achieve approximate solutions with constant approximate ratio arbitrarily close to 1 in near-linear time.

This work has been supported in part by the Zhongguancun Haihua Institute for Frontier Information Technology.

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## Notes

- 1.
\(\tilde{O}(\cdot )\) hides poly-logarithmic factors.

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Duan, R., He, H., Zhang, T. (2020). Near-Linear Time Algorithm for Approximate Minimum Degree Spanning Trees. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham. https://doi.org/10.1007/978-3-030-61792-9_2

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