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Variants of Merge-Width and Applications

Karolina Drabik, Maël Dumas, Colin Geniet, Jakub Nowakowski, Michał Pilipczuk, Szymon Toruńczyk

TL;DR

This work characterisation via definable merge-width uses vertex orderings inspired by generalised colouring numbers from sparsity theory, and enables the first non-trivial approximation algorithm for merge-width, running in time $n^{O(1)} \cdot 2^n$.

Abstract

Merge-width is a recently introduced family of graph parameters that unifies treewidth, clique-width, twin-width, and generalised colouring numbers. We prove the equivalence of several alternative definitions of merge-width, thus demonstrating the robustness of the notion. Our characterisation via definable merge-width uses vertex orderings inspired by generalised colouring numbers from sparsity theory, and enables us to obtain the first non-trivial approximation algorithm for merge-width, running in time $n^{O(1)} \cdot 2^n$. We also obtain a new characterisation of bounded clique-width in terms of vertex orderings, and establish that graphs of bounded merge-width admit sparse quotients with bounded strong colouring numbers, are quasi-isometric to graphs of bounded expansion, and admit neighbourhood covers with constant overlap. We also discuss several other variants of merge-width and connections to adjacency labelling schemes.

Variants of Merge-Width and Applications

TL;DR

This work characterisation via definable merge-width uses vertex orderings inspired by generalised colouring numbers from sparsity theory, and enables the first non-trivial approximation algorithm for merge-width, running in time .

Abstract

Merge-width is a recently introduced family of graph parameters that unifies treewidth, clique-width, twin-width, and generalised colouring numbers. We prove the equivalence of several alternative definitions of merge-width, thus demonstrating the robustness of the notion. Our characterisation via definable merge-width uses vertex orderings inspired by generalised colouring numbers from sparsity theory, and enables us to obtain the first non-trivial approximation algorithm for merge-width, running in time . We also obtain a new characterisation of bounded clique-width in terms of vertex orderings, and establish that graphs of bounded merge-width admit sparse quotients with bounded strong colouring numbers, are quasi-isometric to graphs of bounded expansion, and admit neighbourhood covers with constant overlap. We also discuss several other variants of merge-width and connections to adjacency labelling schemes.
Paper Structure (43 sections, 48 theorems, 105 equations, 4 figures)

This paper contains 43 sections, 48 theorems, 105 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Merge-width
  3. Main result
  4. Applications
  5. Exponential-time approximation of merge-width
  6. Sparse quotients, quasi-isometries, and neighbourhood covers
  7. Other variants of merge-width
  8. Preliminaries
  9. Basic notation and graphs
  10. VC-dimension and neighbourhood complexity
  11. Partitions
  12. Flips and flip metric
  13. Generalised colouring numbers
  14. Twin-width
  15. Merge-width
  16. ...and 28 more sections

Key Result

Theorem 1.1

The following are equivalent for a graph class $\mathscr{C}$:

Figures (4)

  • Figure 1: The tree $T_3$ of \ref{['ex:run-ex']}, with partition $\mathcal{P}_2$ in blue, $\mathcal{P}_3$ in purple and set of resolved pairs $R_3$ in red.
  • Figure 2: Illustration of $4$ differents cases for witnesses of a part $X\in \mathcal{P}_i$. Colored blobs correspond to radius-$2$ balls of the witnesses in $R_{i+1}$, red edges and paths are in $R_{i+1}$.
  • Figure 3: Illustration of the construction of the dual set for $(X',Y')$ with $X',Y'\in \mathcal{P}_i$.
  • Figure 4: Illustration of \ref{['cl:zell']}. The partition $\mathcal{P}'_i$ is in blue. Black edges are from $E(G)$, dashed gray edges are from $E(\overline{G})$, red edges are from $R_{i+1}$.

Theorems & Definitions (106)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1: Sauer-Shelah-Perles lemma
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4: gen-col-nb
  • ...and 96 more