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.
@InProceedings{lee_et_al:LIPIcs.WADS.2025.43, author = {Lee, Pin-Hsian and Tsai, Meng-Tsung and Wang, Hung-Lung}, title = {{On the Complexity of Finding 1-Center Spanning Trees}}, booktitle = {19th International Symposium on Algorithms and Data Structures (WADS 2025)}, pages = {43:1--43:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-398-0}, ISSN = {1868-8969}, year = {2025}, volume = {349}, editor = {Morin, Pat and Oh, Eunjin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.43}, URN = {urn:nbn:de:0030-drops-242743}, doi = {10.4230/LIPIcs.WADS.2025.43}, annote = {Keywords: Treeradius, Spanning tree polytope, Shortest s, t-path polytope} }
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