Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra

Authors Takehiro Ito , Naonori Kakimura , Naoyuki Kamiyama , Yusuke Kobayashi , Shun-ichi Maezawa , Yuta Nozaki , Yoshio Okamoto

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Takehiro Ito
  • Graduate School of Information Sciences, Tohoku University, Japan
Naonori Kakimura
  • Faculty of Science and Technology, Keio University, Yokohama, Japan
Naoyuki Kamiyama
  • Institute of Mathematics for Industry, Kyushu University, Fukouka, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan
Shun-ichi Maezawa
  • Department of Mathematics, Tokyo University of Science, Japan
Yuta Nozaki
  • Faculty of Environment and Information Sciences, Yokohama National University, Japan
  • SKCM, Hiroshima University, Japan
Yoshio Okamoto
  • Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo, Japan


This work was motivated by the talks of Vincent Pilaud and Jean Cardinal at Workshop "Polytope Diameter and Related Topics," held online on September 2, 2022. We thank them for inspiration. We are also grateful to Yuni Iwamasa and Kenta Ozeki for the discussion and to the anonymous reviewers for their helpful comments.

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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 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 82:1-82:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We prove that the computation of a combinatorial shortest path between two vertices of a graph associahedron, introduced by Carr and Devadoss, is NP-hard. This resolves an open problem raised by Cardinal. A graph associahedron is a generalization of the well-known associahedron. The associahedron is obtained as the graph associahedron of a path. It is a tantalizing and important open problem in theoretical computer science whether the computation of a combinatorial shortest path between two vertices of the associahedron can be done in polynomial time, which is identical to the computation of the flip distance between two triangulations of a convex polygon, and the rotation distance between two rooted binary trees. Our result shows that a certain generalized approach to tackling this open problem is not promising. As a corollary of our theorem, we prove that the computation of a combinatorial shortest path between two vertices of a polymatroid base polytope cannot be done in polynomial time unless P = NP. Since a combinatorial shortest path on the matroid base polytope can be computed in polynomial time, our result reveals an unexpected contrast between matroids and polymatroids.

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ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Theory of computation → Problems, reductions and completeness
  • Graph associahedra
  • combinatorial shortest path
  • NP-hardness
  • polymatroids


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  1. Oswin Aichholzer, Wolfgang Mulzer, and Alexander Pilz. Flip distance between triangulations of a simple polygon is NP-complete. Discret. Comput. Geom., 54(2):368-389, 2015. URL: https://doi.org/10.1007/s00454-015-9709-7.
  2. Benjamin Aram Berendsohn. The diameter of caterpillar associahedra. In Artur Czumaj and Qin Xin, editors, 18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022, June 27-29, 2022, Tórshavn, Faroe Islands, volume 227 of LIPIcs, pages 14:1-14:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SWAT.2022.14.
  3. Maike Buchin, Anna Lubiw, Arnaud de Mesmay, and Saul Schleimer. Computation and reconfiguration in low-dimensional topological spaces (Dagstuhl Seminar 22062). Dagstuhl Reports, 12(2):17-66, 2022. URL: https://doi.org/10.4230/DagRep.12.2.17.
  4. Jean Cardinal, Stefan Langerman, and Pablo Pérez-Lantero. On the diameter of tree associahedra. Electron. J. Comb., 25(4):4, 2018. URL: https://doi.org/10.37236/7762.
  5. Jean Cardinal, Arturo I. Merino, and Torsten Mütze. Efficient generation of elimination trees and graph associahedra. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 2128-2140. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.84.
  6. Jean Cardinal, Lionel Pournin, and Mario Valencia-Pabon. Diameter estimates for graph associahedra. Ann. Comb., 26:873-902, 2022. URL: https://doi.org/10.1007/s00026-022-00598-z.
  7. Jean Cardinal, Lionel Pournin, and Mario Valencia-Pabon. The rotation distance of brooms, 2022. URL: https://doi.org/10.48550/arXiv.2211.07984.
  8. 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.
  9. 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.
  10. Sean Cleary and Katherine St. John. Rotation distance is fixed-parameter tractable. Inf. Process. Lett., 109(16):918-922, 2009. URL: https://doi.org/10.1016/j.ipl.2009.04.023.
  11. Sean Cleary and Katherine St. John. A linear-time approximation algorithm for rotation distance. J. Graph Algorithms Appl., 14(2):385-390, 2010. URL: https://doi.org/10.7155/jgaa.00212.
  12. 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.
  13. Jack Edmonds. Submodular functions, matroids, and certain polyhedra. In Richard Guy, Haim Hanani, Norbert Sauer, and Johanan Schönheim, editors, Combinatorial Structures and their Applications, pages 69-87. Gordon and Breach, New York, NY, 1970. Google Scholar
  14. Uriel Feige and Mohammad Mahdian. Finding small balanced separators. In Jon M. Kleinberg, editor, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 375-384. ACM, 2006. URL: https://doi.org/10.1145/1132516.1132573.
  15. Alan M. Frieze and Shang-Hua Teng. On the complexity of computing the diameter of a polytope. Comput. Complex., 4:207-219, 1994. URL: https://doi.org/10.1007/BF01206636.
  16. Satoru Fujishige. Submodular Functions and Optimization, volume 58 of Annals of Discrete Mathematics. Elsevier, Amsterdam, The Netherlands, 2nd edition, 2005. Google Scholar
  17. Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors. Handbook of Discrete and Computational Geometry, Third Edition. CRC Press LLC, 2017. Google Scholar
  18. Takehiro Ito, Erik D. Demaine, Nicholas J. A. Harvey, Christos H. Papadimitriou, Martha Sideri, Ryuhei Uehara, and Yushi Uno. On the complexity of reconfiguration problems. Theor. Comput. Sci., 412(12-14):1054-1065, 2011. URL: https://doi.org/10.1016/j.tcs.2010.12.005.
  19. Volker Kaibel and Marc E. Pfetsch. Some algorithmic problems in polytope theory. In Michael Joswig and Nobuki Takayama, editors, Algebra, Geometry, and Software Systems [outcome of a Dagstuhl seminar], pages 23-47. Springer, 2003. URL: https://doi.org/10.1007/978-3-662-05148-1_2.
  20. Iyad A. Kanj, Eric Sedgwick, and Ge Xia. Computing the flip distance between triangulations. Discret. Comput. Geom., 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 Petra Berenbrink, Patricia Bouyer, Anuj Dawar, and Mamadou Moustapha Kanté, editors, 40th International Symposium on Theoretical Aspects of Computer Science, STACS 2023, March 7-9, 2023, Hamburg, Germany, volume 254 of LIPIcs, pages 44:1-44:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.STACS.2023.44.
  22. Anna Lubiw and Vinayak Pathak. Flip distance between two triangulations of a point set is NP-complete. Comput. Geom., 49:17-23, 2015. URL: https://doi.org/10.1016/j.comgeo.2014.11.001.
  23. Joan M. Lucas. An improved kernel size for rotation distance in binary trees. Inf. Process. Lett., 110(12-13):481-484, 2010. URL: https://doi.org/10.1016/j.ipl.2010.04.022.
  24. Thibault Manneville and Vincent Pilaud. Graph properties of graph associahedra. Seminaire Lotharingien de Combinatoire, 73:B73d, 2015. Google Scholar
  25. Alexander Pilz. Flip distance between triangulations of a planar point set is APX-hard. Comput. Geom., 47(5):589-604, 2014. URL: https://doi.org/10.1016/j.comgeo.2014.01.001.
  26. Alex Postnikov, Victor Reiner, and Lauren Williams. Faces of generalized permutohedra. Doc. Math., 13:207-273, 2008. Google Scholar
  27. Lionel Pournin. The diameter of associahedra. Advances in Mathematics, 259:13-42, 2014. URL: https://doi.org/10.1016/j.aim.2014.02.035.
  28. 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.
  29. Laura Sanità. The diameter of the fractional matching polytope and its hardness implications. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 910-921. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00090.
  30. Daniel D. Sleator, Robert E. Tarjan, and William P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc., 1:647-681, 1988. URL: https://doi.org/10.1090/S0894-0347-1988-0928904-4.
  31. Donald M. Topkis. Adjacency on polymatroids. Math. Program., 30(2):229-237, 1984. URL: https://doi.org/10.1007/BF02591887.