License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2023.82
URN: urn:nbn:de:0030-drops-181344
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18134/
Ito, Takehiro ;
Kakimura, Naonori ;
Kamiyama, Naoyuki ;
Kobayashi, Yusuke ;
Maezawa, Shun-ichi ;
Nozaki, Yuta ;
Okamoto, Yoshio
Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra
Abstract
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.
BibTeX - Entry
@InProceedings{ito_et_al:LIPIcs.ICALP.2023.82,
author = {Ito, Takehiro and Kakimura, Naonori and Kamiyama, Naoyuki and Kobayashi, Yusuke and Maezawa, Shun-ichi and Nozaki, Yuta and Okamoto, Yoshio},
title = {{Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {82:1--82:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-278-5},
ISSN = {1868-8969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18134},
URN = {urn:nbn:de:0030-drops-181344},
doi = {10.4230/LIPIcs.ICALP.2023.82},
annote = {Keywords: Graph associahedra, combinatorial shortest path, NP-hardness, polymatroids}
}
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Keywords: |
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Graph associahedra, combinatorial shortest path, NP-hardness, polymatroids |
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Collection: |
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50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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05.07.2023 |
