The price of homogeneity is polynomial

2026-02-02

AI Summary

This work addresses the exponential parameter dependence arising from the Homogeneous Wall Lemma in graph minor algorithms by investigating the growth rate of its central function \( h(q,k) \). By integrating tools from graph minor theory, structural graph theory, and combinatorial analysis, the authors provide a refined characterization of coloring and subgraph isomorphism properties of wall structures. They establish, for the first time, that \( h(q,k) \in O(q^4 k^6) \), thereby reducing the previously known exponential bound in \( q \) to a polynomial one. This result resolves an open problem posed by Sau et al. at ICALP 2020 and significantly improves the parameterized complexity of graph algorithms based on the irrelevant vertex technique, enhancing their practical applicability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We provide explicit and polynomial bounds for the Homogeneous Wall Lemma which occurred for the first time implicitly in the $13$th entry of Robertson and Seymour's Graph Minors Series [JCTB 1990] and has since become a cornerstone in the algorithmic theory of graph minors. A wall where each brick is assigned a set of colours is said to be homogeneous if each brick is assigned the same set of colours. The Homogeneous Wall Lemma says that there exists a function $h$ that, given non-negative integers $q$ and $k$ and an $h(q,k)$-wall $W$ where each brick is assigned a, possibly empty, subset of $\{ 1, \ldots , q \}$ contains a $k$-wall $W'$ as a subgraph such that, if one assigns to each brick $B$ of $W'$ the union of the sets assigned to the bricks of $W$ in its interior, then $W'$ is homogeneous. It is well-known that $h(q,k) \in k^{\mathcal{O}(q)}$. The Homogeneous Wall Lemma plays a key role in most applications of the Irrelevant Vertex Technique where an exponential dependency of $h$ on $q$ usually causes non-uniform dependencies on meta-parameters at best and additional exponential blow-ups at worst. By proving that $h(q,k) \in \mathcal{O}(q^4 \cdot k^6)$, we provide a positive answer to a problem raised by Sau, Stamoulis, and Thilikos [ICALP 2020].

Problem

• Homogeneous Wall Lemma
• graph minors
• Irrelevant Vertex Technique
• polynomial bounds
• parameterized complexity

Innovation

• Homogeneous Wall Lemma
• polynomial bound
• graph minors
• Irrelevant Vertex Technique
• parameterized algorithms