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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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Subject Classification

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.

Cite As Get BibTex

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.

References

  1. Amir Abboud, MohammadHossein Bateni, Vincent Cohen-Addad, Karthik C. S., and Saeed Seddighin. On complexity of 1-center in various metrics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), volume 275 of LIPIcs, pages 1:1-1:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.1.
  2. Amir Abboud, Nick Fischer, Elazar Goldenberg, Karthik C. S., and Ron Safier. Can you solve closest string faster than exhaustive search? In 31st Annual European Symposium on Algorithms (ESA), volume 274 of LIPIcs, pages 3:1-3:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.3.
  3. Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 377-391. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH28.
  4. Lowell W. Beineke, Robin J. Wilson, Ortrud R. Oellermann, Dirk Meierling, Lutz Volkmann, Matthias Kriesell, Kiyoshi Ando, R. J. Faudree, Michael Ferrara, Ronald J. Gould, Dieter Rautenbach, Bruce Reed, Ian Anderson, Keith Edwards, Graham Farr, Béla Bollobás, Oliver Riordan, F. T. Boesch, A. Satyanarayana, C. L. Suffel, Abdol-Hossein Esfahanian, R. A. Bailey, and Peter J. Cameron, editors. Topics in Structural Graph Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, UK, 2012. Google Scholar
  5. Amir Ben-Dor, Giuseppe Lancia, Jennifer Perone, and R. Ravi. Banishing bias from consensus sequences. In Combinatorial Pattern Matching, 8th Annual Symposium (CPM), volume 1264 of Lecture Notes in Computer Science, pages 247-261. Springer, 1997. URL: https://doi.org/10.1007/3-540-63220-4_63.
  6. Piotr Berman and Toshihiro Fujito. On approximation properties of the independent set problem for degree 3 graphs. In Algorithms and Data Structures, pages 449-460, Berlin, Heidelberg, 1995. Springer Berlin Heidelberg. Google Scholar
  7. Paul S. Bonsma. Spanning trees with many leaves in graphs with minimum degree three. SIAM J. Discret. Math., 22(3):920-937, 2008. URL: https://doi.org/10.1137/060664318.
  8. Hajo Broersma and Xueliang Li. Spanning trees with many or few colors in edge-colored graphs. Discuss. Math. Graph Theory, 17(2):259-269, 1997. URL: https://doi.org/10.7151/DMGT.1053.
  9. R. L. Brooks. On colouring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society, 37(2):194-197, April 1941. Google Scholar
  10. Yi-Jun Chang, Martin Farach-Colton, Tsan-sheng Hsu, and Meng-Tsung Tsai. Streaming complexity of spanning tree computation. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 154 of LIPIcs, pages 34:1-34:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.STACS.2020.34.
  11. Miroslav Chlebík and Janka Chlebíková. Inapproximability results for bounded variants of optimization problems. In Fundamentals of Computation Theory, 14th International Symposium (FCT), volume 2751 of Lecture Notes in Computer Science, pages 27-38. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-45077-1_4.
  12. Moti Frances and Ami Litman. On covering problems of codes. Theory Comput. Syst., 30(2):113-119, 1997. URL: https://doi.org/10.1007/S002240000044.
  13. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  14. Leszek Gasieniec, Jesper Jansson, and Andrzej Lingas. Efficient approximation algorithms for the hamming center problem. In Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 905-906. ACM/SIAM, 1999. URL: http://dl.acm.org/citation.cfm?id=314500.315081.
  15. R. E. Gomory and T. C. Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551-570, 1961. Google Scholar
  16. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/JCSS.2000.1727.
  17. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/JCSS.2001.1774.
  18. David S. Johnson and Mario Szegedy. What are the least tractable instances of max independent set? In Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '99, pages 927-928, USA, 1999. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=314500.315093.
  19. Daniel J. Kleitman and Douglas B. West. Spanning trees with many leaves. SIAM J. Discret. Math., 4(1):99-106, 1991. URL: https://doi.org/10.1137/0404010.
  20. J. Kevin Lanctôt, Ming Li, Bin Ma, Shaojiu Wang, and Louxin Zhang. Distinguishing string selection problems. Inf. Comput., 185(1):41-55, 2003. URL: https://doi.org/10.1016/S0890-5401(03)00057-9.
  21. Ming Li, Bin Ma, and Lusheng Wang. On the closest string and substring problems. J. ACM, 49(2):157-171, 2002. URL: https://doi.org/10.1145/506147.506150.
  22. Petr Lisonek. New maximal two-distance sets. J. Comb. Theory A, 77(2):318-338, 1997. URL: https://doi.org/10.1006/JCTA.1997.2749.
  23. Christos H. Papadimitriou and Kenneth Steiglitz. Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, 1982. Google Scholar
  24. Nikola Stojanovic, Piotr Berman, Deborah Gumucio, Ross C. Hardison, and Webb Miller. A linear-time algorithm for the 1-mismatch problem. In Algorithms and Data Structures, 5th International Workshop (WADS), volume 1272 of Lecture Notes in Computer Science, pages 126-135. Springer, 1997. URL: https://doi.org/10.1007/3-540-63307-3_53.
  25. Bang Ye Wu and Kun-Mao Chao. Spanning trees and optimization problems. Discrete mathematics and its applications. Chapman & Hall, 2004. Google Scholar
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