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Computing Distances on Graph Associahedra Is Fixed-Parameter Tractable

Authors Luís Felipe I. Cunha , Ignasi Sau , Uéverton S. Souza , Mario Valencia-Pabon

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ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Combinatorics
Keywords
  • graph associahedra
  • elimination tree
  • rotation distance
  • parameterized complexity
  • fixed-parameter tractable algorithm
  • combinatorial shortest path
  • reconfiguration

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Abstract

An elimination tree of a connected graph G is a rooted tree on the vertices of G obtained by choosing a root v and recursing on the connected components of G-v to obtain the subtrees of v. The graph associahedron of G is a polytope whose vertices correspond to elimination trees of G and whose edges correspond to tree rotations, a natural operation between elimination trees. These objects generalize associahedra, which correspond to the case where G is a path. Ito et al. [ICALP 2023] recently proved that the problem of computing distances on graph associahedra is NP-hard. In this paper we prove that the problem, for a general graph G, is fixed-parameter tractable parameterized by the distance k. Prior to our work, only the case where G is a path was known to be fixed-parameter tractable. To prove our result, we use a novel approach based on a marking scheme that restricts the search to a set of vertices whose size is bounded by a (large) function of k.

Cite As Get BibTex

Luís Felipe I. Cunha, Ignasi Sau, Uéverton S. Souza, and Mario Valencia-Pabon. Computing Distances on Graph Associahedra Is Fixed-Parameter Tractable. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 63:1-63:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.63

Author Details

Luís Felipe I. Cunha
  • Instituto de Computação, Universidade Federal Fluminense, Brasil
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France
Uéverton S. Souza
  • Instituto de Computação, Universidade Federal Fluminense, Brasil
  • IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brasil
Mario Valencia-Pabon
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

Funding

  • Cunha, Luís Felipe I.: FAPERJ-JCNE (E-26/201.372/2022) and CNPq-Universal (406173/ 2021-4).
  • Sau, Ignasi: French project ELiT (ANR-20-CE48-0008-01).
  • Souza, Uéverton S.: CNPq (312344/2023-6), and FAPERJ (E-26/201.344/2021).
  • Valencia-Pabon, Mario: French project Abysm (ANR-23-CE48-0017).

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