The price of homogeneity is polynomial

TL;DR

This paper establishes explicit polynomial bounds for the Homogeneous Wall Lemma, a fundamental result in graph minor theory, significantly improving previous exponential bounds and enabling more efficient algorithmic applications.

Contribution

It proves that the function h(q,k) can be bounded polynomially as O(q^4 · k^6), resolving an open problem and enhancing the lemma's applicability.

Findings

Established polynomial bound h(q,k) = O(q^4 · k^6)

Improved from previous exponential bounds

Facilitates more efficient graph minor algorithms

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].