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Computing Distances on Graph Associahedra Is Fixed-Parameter Tractable

Authors Luís Felipe I. Cunha , Ignasi Sau , Uéverton S. Souza , Mario Valencia-Pabon

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LIPIcs.ICALP.2025.63.pdf
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ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Combinatorics
Keywords
  • graph associahedra
  • elimination tree
  • rotation distance
  • parameterized complexity
  • fixed-parameter tractable algorithm
  • combinatorial shortest path
  • reconfiguration

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Abstract

An elimination tree of a connected graph G is a rooted tree on the vertices of G obtained by choosing a root v and recursing on the connected components of G-v to obtain the subtrees of v. The graph associahedron of G is a polytope whose vertices correspond to elimination trees of G and whose edges correspond to tree rotations, a natural operation between elimination trees. These objects generalize associahedra, which correspond to the case where G is a path. Ito et al. [ICALP 2023] recently proved that the problem of computing distances on graph associahedra is NP-hard. In this paper we prove that the problem, for a general graph G, is fixed-parameter tractable parameterized by the distance k. Prior to our work, only the case where G is a path was known to be fixed-parameter tractable. To prove our result, we use a novel approach based on a marking scheme that restricts the search to a set of vertices whose size is bounded by a (large) function of k.

Cite As Get BibTex

Luís Felipe I. Cunha, Ignasi Sau, Uéverton S. Souza, and Mario Valencia-Pabon. Computing Distances on Graph Associahedra Is Fixed-Parameter Tractable. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 63:1-63:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.63

Author Details

Luís Felipe I. Cunha
  • Instituto de Computação, Universidade Federal Fluminense, Brasil
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France
Uéverton S. Souza
  • Instituto de Computação, Universidade Federal Fluminense, Brasil
  • IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brasil
Mario Valencia-Pabon
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

Funding

  • Cunha, Luís Felipe I.: FAPERJ-JCNE (E-26/201.372/2022) and CNPq-Universal (406173/ 2021-4).
  • Sau, Ignasi: French project ELiT (ANR-20-CE48-0008-01).
  • Souza, Uéverton S.: CNPq (312344/2023-6), and FAPERJ (E-26/201.344/2021).
  • Valencia-Pabon, Mario: French project Abysm (ANR-23-CE48-0017).

References

  1. Georgy M. Adel’son-Vel’skii and Evgenii M. Landis. An algorithm for the organization of information. Soviet Mathematics Doklady, 3:1259-1263, 1962. URL: https://zhjwpku.com/assets/pdf/AED2-10-avl-paper.pdf.
  2. Oswin Aichholzer, Wolfgang Mulzer, and Alexander Pilz. Flip Distance Between Triangulations of a Simple Polygon is NP-Complete. Discrete & Computational Geometry, 54(2):368-389, 2015. URL: https://doi.org/10.1007/S00454-015-9709-7.
  3. Benjamin Aram Berendsohn. The diameter of caterpillar associahedra. In Proc. of the 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), volume 227 of LIPIcs, pages 14:1-14:12, 2022. URL: https://doi.org/10.4230/LIPICS.SWAT.2022.14.
  4. Nicolas Bousquet, Amer E. Mouawad, Naomi Nishimura, and Sebastian Siebertz. A survey on the parameterized complexity of reconfiguration problems. Computer Science Review, 53:100663, 2024. URL: https://doi.org/10.1016/J.COSREV.2024.100663.
  5. Jean Cardinal, Linda Kleist, Boris Klemz, Anna Lubiw, Torsten Mütze, Alexander Neuhaus, and Lionel Pournin. Working grup 4.2: Rotation distance between elimination trees. Report from Dagstuhl Seminar 22062: Computation and Reconfiguration in Low-Dimensional Topological Spaces, page 35, 2022. URL: https://drops.dagstuhl.de/storage/04dagstuhl-reports/volume12/issue02/22062/DagRep.12.2.17/DagRep.12.2.17.pdf.
  6. Jean Cardinal, Stefan Langerman, and Pablo Pérez-Lantero. On the diameter of tree associahedra. Electronic Journal of Combinatorics, 25(4):4, 2018. URL: https://doi.org/10.37236/7762.
  7. Jean Cardinal, Arturo Merino, and Torsten Mütze. Efficient generation of elimination trees and graph associahedra. In Proc. of the 33rd ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2128-2140, 2022. URL: https://doi.org/10.1137/1.9781611977073.84.
  8. Jean Cardinal, Lionel Pournin, and Mario Valencia-Pabon. Diameter estimates for graph associahedra. Annals of Combinatorics, 26:873-902, 2022. URL: https://doi.org/10.1007/s00026-022-00598-z.
  9. Jean Cardinal, Lionel Pournin, and Mario Valencia-Pabon. The rotation distance of brooms. European Journal of Combinatorics, 118:103877, 2024. URL: https://doi.org/10.1016/J.EJC.2023.103877.
  10. Jean Cardinal and Raphael Steiner. Shortest paths on polymatroids and hypergraphic polytopes. CoRR, abs/2311.00779, 2023. URL: https://doi.org/10.48550/arXiv.2311.00779.
  11. Michael Carr and Satyan L. Devadoss. Coxeter complexes and graph-associahedra. Topology and its Applications, 153(12):2155-2168, 2006. URL: https://doi.org/10.1016/j.topol.2005.08.010.
  12. Cesar Ceballos, Francisco Santos, and Günter M. Ziegler. Many non-equivalent realizations of the associahedron. Combinatorica, 35(5):513-551, 2015. URL: https://doi.org/10.1007/s00493-014-2959-9.
  13. Sean Cleary. Restricted rotation distance between k-ary trees. Journal of Graph Algorithms and Applications, 27(1):19-33, 2023. URL: https://doi.org/10.7155/JGAA.00611.
  14. Sean Cleary and Katherine St. John. Rotation distance is fixed-parameter tractable. Information Processing Letters, 109(16):918-922, 2009. URL: https://doi.org/10.1016/J.IPL.2009.04.023.
  15. Sean Cleary and Katherine St. John. A linear-time approximation algorithm for rotation distance. Journal of Graph Algorithms and Applications, 14(2):385-390, 2010. URL: https://doi.org/10.7155/JGAA.00212.
  16. Satyan L. Devadoss. A realization of graph associahedra. Discrete Mathematics, 309(1):271-276, 2009. URL: https://doi.org/10.1016/J.DISC.2007.12.092.
  17. Leonidas J. Guibas and Robert Sedgewick. A dichromatic framework for balanced trees. In Proc. of the 19th Annual Symposium on Foundations of Computer Science (FOCS), pages 8-21, 1978. URL: https://doi.org/10.1109/SFCS.1978.3.
  18. Christophe Hohlweg and Carsten E. M. C. Lange. Realizations of the associahedron and cyclohedron. Discrete & Computational Geometry, 37(4):517-543, 2007. URL: https://doi.org/10.1007/S00454-007-1319-6.
  19. Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, and Yoshio Okamoto. Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra. In Prof. of the 50th International Colloquium on Automata, Languages, and Programming (ICALP), volume 261 of LIPIcs, pages 82:1-82:17, 2023. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.82.
  20. Iyad Kanj, Eric Sedgwick, and Ge Xia. Computing the flip distance between triangulations. Discrete & Computational Geometry, 58(2):313-344, 2017. URL: https://doi.org/10.1007/S00454-017-9867-X.
  21. Haohong Li and Ge Xia. An O(3.82^k) time FPT algorithm for convex flip distance. In Proc. of the 40th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 254 of LIPIcs, pages 44:1-44:14, 2023. URL: https://doi.org/10.4230/LIPICS.STACS.2023.44.
  22. Jean-Louis Loday. Realization of the Stasheff polytope. Archiv der Mathematik, 83:267-278, 2004. URL: https://doi.org/10.1007/s00013-004-1026-y.
  23. Anna Lubiw and Vinayak Pathak. Flip distance between two triangulations of a point set is NP-complete. Computational Geometry, 49:17-23, 2015. URL: https://doi.org/10.1016/J.COMGEO.2014.11.001.
  24. Joan M. Lucas. An improved kernel size for rotation distance in binary trees. Information Processing Letters, 110(12-13):481-484, 2010. URL: https://doi.org/10.1016/J.IPL.2010.04.022.
  25. Thibault Manneville and Vincent Pilaud. Graph properties of graph associahedra. CoRR, abs/1409.8114, 2010. URL: https://arxiv.org/abs/1409.8114.
  26. Jaroslav Nesetril and Patrice Ossona de Mendez. Sparsity - Graphs, Structures, and Algorithms, volume 28 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-27875-4.
  27. Alexander Pilz. Flip distance between triangulations of a planar point set is APX-hard. Computational Geometry, 47(5):589-604, 2014. URL: https://doi.org/10.1016/J.COMGEO.2014.01.001.
  28. Alex Postnikov, Victor Reiner, and Lauren Williams. Faces of generalized permutohedra. Documenta Mathematica, 13:207-273, 2008. URL: https://doi.org/10.4171/DM/248.
  29. Alexander Postnikov. Permutohedra, associahedra, and beyond. International Mathematics Research Notices, 2009(6):1026-1106, 2009. URL: https://doi.org/10.1093/imrn/rnn153.
  30. Lionel Pournin. The diameter of associahedra. Advances in Mathematics, 259:13-42, 2014. URL: https://doi.org/10.1016/j.aim.2014.02.035.
  31. Lionel Pournin. The asymptotic diameter of cyclohedra. Israel Journal of Mathematics, 219(2):609-635, 2017. URL: https://doi.org/10.1007/s11856-017-1492-0.
  32. Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. Journal of the American Mathematical Society, 1(3):647-681, 1988. URL: https://doi.org/10.1090/S0894-0347-1988-0928904-4.
  33. Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652-686, 1985. URL: https://doi.org/10.1145/3828.3835.
  34. Richard P. Stanley. Catalan numbers. Cambridge University Press, 2015. URL: https://doi.org/10.1017/CBO9781139871495.
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