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The price of homogeneity is polynomial

Maximilian Gorsky, Michał T. Seweryn, Sebastian Wiederrecht

TL;DR

This work resolves a long-standing bottleneck in the algorithmic application of Robertson–Seymour Graph Minors theory by providing explicit polynomial bounds for the Homogeneous Wall Lemma. The authors introduce a polynomial homogeneity framework based on planar-like meshes, strips, tiles, and padding, and prove that from a large colorfully-witnessed wall one can extract a homogeneous flat subwall whose tangle is a truncation of the original, computable in poly-time. The key contribution is showing $h(q,k) bond ilde{O}(q^4 \, k^6)$, replacing the prior $k^{ ilde{O}(q)}$-type dependencies and thereby removing exponential blow-ups in many Irrelevant Vertex–based algorithms. This improvement propagates to major parameterized problems such as $k$-H-Minor Deletion and related minor-closed graph class algorithms, reducing non-uniformities and enabling more scalable, uniform running times in practice. The results open the path toward stronger polynomially-bounded variants of broader structure theorems (e.g., AGMST) and deepen the understanding of homogenisation in graph minor theory.

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].

The price of homogeneity is polynomial

TL;DR

This work resolves a long-standing bottleneck in the algorithmic application of Robertson–Seymour Graph Minors theory by providing explicit polynomial bounds for the Homogeneous Wall Lemma. The authors introduce a polynomial homogeneity framework based on planar-like meshes, strips, tiles, and padding, and prove that from a large colorfully-witnessed wall one can extract a homogeneous flat subwall whose tangle is a truncation of the original, computable in poly-time. The key contribution is showing , replacing the prior -type dependencies and thereby removing exponential blow-ups in many Irrelevant Vertex–based algorithms. This improvement propagates to major parameterized problems such as -H-Minor Deletion and related minor-closed graph class algorithms, reducing non-uniformities and enabling more scalable, uniform running times in practice. The results open the path toward stronger polynomially-bounded variants of broader structure theorems (e.g., AGMST) and deepen the understanding of homogenisation in graph minor theory.

Abstract

We provide explicit and polynomial bounds for the Homogeneous Wall Lemma which occurred for the first time implicitly in the 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 that, given non-negative integers and and an -wall where each brick is assigned a, possibly empty, subset of contains a -wall as a subgraph such that, if one assigns to each brick of the union of the sets assigned to the bricks of in its interior, then is homogeneous. It is well-known that . The Homogeneous Wall Lemma plays a key role in most applications of the Irrelevant Vertex Technique where an exponential dependency of on usually causes non-uniform dependencies on meta-parameters at best and additional exponential blow-ups at worst. By proving that , we provide a positive answer to a problem raised by Sau, Stamoulis, and Thilikos [ICALP 2020].
Paper Structure (53 sections, 9 theorems, 19 equations, 18 figures)

This paper contains 53 sections, 9 theorems, 19 equations, 18 figures.

Table of Contents

  1. Introduction
  2. The price of homogeneity.
  3. Our result.
  4. A polynomial homogeneity lemma.
  5. Two examples of how to adapt thm:IntroHomoMuralis.
  6. Algorithmic consequences of our result.
  7. A short overview of our proof.
  8. Strips.
  9. Step 1: Homogenise strips.
  10. Step 2: Tiles and further homogenisation.
  11. Step 3: Constructing the homogeneous mesh.
  12. Preliminaries
  13. Tangles
  14. Walls and meshes
  15. Grids and Walls.
  16. ...and 38 more sections

Key Result

Theorem 1.1

There exists a function $f\colon \mathbb{N}^2\to\mathbb{N}$ with $f(q,k) \in \mathcal{O}(q^4 \cdot k^6)$ such that for all non-negative integers $q$ and $k$ and every $q$-colorful graph $G$ with a flat $f(q,k)$-wall $W_0$ there exists a flat $k$-wall $W_1\subseteq W_0$ such that the tangle of $W_1$

Figures (18)

  • Figure 1: A $4$-wall (on the left) and a $(6 \times 6)$-mesh (on the right).
  • Figure 2: Diagrams of (i) a strip in a mesh -- the grey area is the rectangle $\Delta$ and (ii) a "padded" strip partitioned into a middle part and two buffer zones (one of the left and the other on the right).
  • Figure 3: Diagrams of (i) tiles created by overlaying horizontal and vertical strips, (ii) dangerous "end tiles" that are not fully surrounded by buffer zones, and (iii) a submesh cropped to the union of horizontal and vertical strips.
  • Figure 4: Diagrams of (i) a mesh with a rainbow middle row and (ii) the construction of a homogeneous mesh from a rainbow-middle-row mesh.
  • Figure 5: The elementary $4$-wall.
  • ...and 13 more figures

Theorems & Definitions (16)

  • Theorem 1.1
  • Proposition 1.2: Morelle, Sau, Stamoulis, Thilikos MorelleSST2023Faster
  • Corollary 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.2
  • ...and 6 more