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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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Author Details

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.

Subject Classification

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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