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Computing Distances on Graph Associahedra is Fixed-parameter Tractable

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

TL;DR

This paper proves that the problem of computing distances on graph associahedra is fixed-parameter tractable parameterized by the distance $k$ and restricts the search to a set of vertices whose size is bounded by a (large) function of $k$.

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 $k$.

Computing Distances on Graph Associahedra is Fixed-parameter Tractable

TL;DR

This paper proves that the problem of computing distances on graph associahedra is fixed-parameter tractable parameterized by the distance and restricts the search to a set of vertices whose size is bounded by a (large) function of .

Abstract

An elimination tree of a connected graph is a rooted tree on the vertices of obtained by choosing a root and recursing on the connected components of to obtain the subtrees of . The graph associahedron of is a polytope whose vertices correspond to elimination trees of and whose edges correspond to tree rotations, a natural operation between elimination trees. These objects generalize associahedra, which correspond to the case where 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 , is fixed-parameter tractable parameterized by the distance . Prior to our work, only the case where 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 .

Paper Structure

This paper contains 6 sections, 2 theorems, 1 equation, 2 figures.

Table of Contents

  1. Introduction
  2. Overview of the main ideas of the algorithm
  3. Preliminaries
  4. Formal description of the FPT algorithm
  5. Good and bad vertices
  6. Restricting the rotations to small balls around bad vertices

Key Result

Theorem 1

The Rotation Distance problem can be solved in time $f(k) \cdot |V(G)|$, with

Figures (2)

  • Figure 1: A graph $G$ and two of its elimination trees $T$ and $T'$, where the second one is obtained from the first one by the rotation of edge $uv$ (in red).
  • Figure 2: On the left: An elimination tree $T$ of a graph $G$ with adjacent vertices $u$ and $v$. Vertex $v$ has four subtrees, and two of them, namely $T_2$ and $T_3$, contain vertices adjacent to vertex $u$ in $G$. On the right: Elimination tree resulting from $T$ by applying the rotation of $uv$. Since both $G[V(T_2) \cup \{u\}]$ and $G[V(T_3) \cup \{u\}]$ are connected, $T_2$ and $T_3$ become subtrees of $u$ in ${\sf rot}\xspace(T,uv)$.

Theorems & Definitions (5)

  • Theorem 1
  • Definition 2: rotation operation
  • Definition 4: bad vertices
  • Definition 6: union of balls of children-bad vertices
  • Lemma 7