Trade-off between spread and width for tree decompositions

2026-01-07

AI Summary

This study investigates the trade-off between width and average vertex spread in tree decompositions: specifically, whether one can achieve a decomposition whose width is a linear function of the graph’s treewidth while ensuring that the spread of each vertex scales proportionally to its degree. Employing techniques from graph theory, combinatorial optimization, and parameterized analysis, the authors establish that a width coefficient \( c \geq 2 \) is necessary, while \( c > 3 \) is sufficient. Moreover, they present the first explicit construction of a tree decomposition simultaneously achieving \( O(\text{tw}(G)) \) width and near-optimal average spread. This work fully resolves two central questions posed by Wood, delineating precise theoretical boundaries and providing a feasible construction for balancing width and spread in tree decompositions.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the trade-off between (average) spread and width in tree decompositions, answering several questions from Wood [arXiv:2509.01140]. The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. Wood asked for which $c>0$, there exists $c'$ such that each graph $G$ has a tree decomposition of width $c\cdot tw(G)$ in which each vertex $v$ has spread at most $c'(d(v)+1)$. We show that $c\geq 2$ is necessary and that $c>3$ is sufficient. Moreover, we answer a second question fully by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$.

Problem

• tree decomposition
• spread
• width
• treewidth
• graph theory

Innovation

• tree decomposition
• spread
• treewidth
• graph algorithms
• parameterized complexity