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On the Complexity of Finding 1-Center Spanning Trees

Authors Pin-Hsian Lee , Meng-Tsung Tsai , Hung-Lung Wang

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
Keywords
  • Treeradius
  • Spanning tree polytope
  • Shortest s
  • t-path polytope

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Abstract

We consider the problem of finding a spanning tree T of a given undirected graph G such that any other spanning tree can be obtained from T by removing k edges and subsequently adding k edges, where k is minimized over all spanning trees of G. We refer to this minimum k as the treeradius of G. 
Treeradius is an interesting graph parameter with natural interpretations: (1) It is the smallest radius of a Hamming ball centered at an extreme point of the spanning tree polytope that covers the entire polytope. (2) Any graph with bounded treeradius also has bounded treewidth. Consequently, if a problem admits a fixed-parameter algorithm parameterized by treewidth, it also admits a fixed-parameter algorithm parameterized by treeradius. 
In this paper, we show that computing the exact treeradius for n-vertex graphs requires 2^Ω(n) time under the Exponential Time Hypothesis (ETH) and does not admit a PTAS, with an inapproximability bound of 1153/1152, unless P = NP. This hardness result is surprising, as treeradius has significantly higher ETH complexity compared to analogous problems on shortest path polytopes and star subgraph polytopes.

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Pin-Hsian Lee, Meng-Tsung Tsai, and Hung-Lung Wang. On the Complexity of Finding 1-Center Spanning Trees. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.43

Author Details

Pin-Hsian Lee
  • National Taiwan Normal University, Taipei, Taiwan
Meng-Tsung Tsai
  • Academia Sinica, Taipei, Taiwan
Hung-Lung Wang
  • National Taiwan Normal University, Taipei, Taiwan

Funding

This research was supported in part by the National Science and Technology Council under contract NSTC 113-2628-E-003-001-MY3 and 113-2221-E-001-026.

Acknowledgements

We want to thank the anonymous reviewers for their valuable comments.

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