Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra

Media type: Text; E-Article; Electronic Conference Proceeding

Title: Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra

Contributor: Ito, Takehiro [Author]; Kakimura, Naonori [Author]; Kamiyama, Naoyuki [Author]; Kobayashi, Yusuke [Author]; Maezawa, Shun-ichi [Author]; Nozaki, Yuta [Author]; Okamoto, Yoshio [Author]

imprint: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2023

Language: English

DOI: https://doi.org/10.4230/LIPIcs.ICALP.2023.82

Keywords: combinatorial shortest path ; polymatroids ; Graph associahedra ; NP-hardness

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Description: 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.

Access State: Open Access

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