2026-07-22T19:33:53Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:18134 2024-03-06T11:01:06Z ddc:004 open_access

Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra Ito, Takehiro Kakimura, Naonori Kamiyama, Naoyuki Kobayashi, Yusuke Maezawa, Shun-ichi Nozaki, Yuta Okamoto, Yoshio Graph associahedra combinatorial shortest path NP-hardness polymatroids 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Takehiro Ito and Naonori Kakimura and Naoyuki Kamiyama and Yusuke Kobayashi and Shun-ichi Maezawa and Yuta Nozaki and Yoshio Okamoto 2023 Is Part Of LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2023.82 urn:nbn:de:0030-drops-181344 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.82 eng https://creativecommons.org/licenses/by/4.0/legalcode