Computing Distances on Graph Associahedra is Fixed-parameter Tractable

2025-04-25

AI Summary

This paper addresses the problem of computing the shortest-path distance between two elimination trees on a graph associahedron—a problem known to be NP-hard for general graphs and previously admitting an FPT algorithm only for path graphs. We establish, for the first time, that this distance computation is fixed-parameter tractable (FPT) with respect to the target distance $k$. Our method introduces a combinatorial labeling framework coupled with bounded search: we model tree transformations via rotations and compress vertex-set representations to confine the search space within a finite bound dependent solely on $k$. The resulting algorithm runs in time $f(k) cdot mathrm{poly}(n)$, where $f$ is a superpolynomial function. This result overcomes prior structural restrictions—limited to paths—and elevates the problem from NP-hardness to FPT for arbitrary graphs, significantly broadening the scope of efficiently solvable instances.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $k$.

Problem

• Computing distances on graph associahedra is NP-hard
• Proving fixed-parameter tractability for general graphs
• Using a marking scheme to bound search space

Innovation

• Fixed-parameter tractable distance computation
• Marking scheme restricts search space
• Generalizes path-based associahedra results