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Dynamic Meta-Kernelization

Christian Bertram, Deborah Haun, Mads Vestergaard Jensen, Tuukka Korhonen

TL;DR

This work advances dynamic kernelization by introducing a dynamic protrusion-decomposition framework for H-topological-minor-free graphs, enabling maintenance of approximately optimal treewidth-modulators and efficient protrusion replacements under edge and vertex updates. The core idea is to keep a protrusion decomposition whose root bag tracks a modulator, and to replace large protrusions in one shot via a protrusion replacement automaton, all while updating in amortized O(log |G|) time. The authors prove a meta-theorem: for CMSO$_2$-definable problems on CMSO-definable minor- or apex-minor-free classes that are linearly treewidth-bounding with FII, there exist dynamic kernelization data structures producing kernels of size O(OPT(G)) that can be maintained with constant or logarithmic per-update work, and they show direct consequences for dynamic approximation and dynamic FPT algorithms. The framework integrates dynamic treewidth techniques with advanced protrusion-balancing methods, yielding a modular approach that extends static kernelization results to dynamic settings and opens avenues for dynamic preprocessing in sparse graphs. Overall, the paper provides a principled, scalable path to dynamic preprocessing and kernelization, with specific gains for Dominating Set on planar graphs and a broad class of CMSO$_2$ problems on minor- and topological-minor-free graphs.

Abstract

Kernelization studies polynomial-time preprocessing algorithms. Over the last 20 years, the most celebrated positive results of the field have been linear kernels for classical NP-hard graph problems on sparse graph classes. In this paper, we lift these results to the dynamic setting. As the canonical example, Alber, Fellows, and Niedermeier [J. ACM 2004] gave a linear kernel for dominating set on planar graphs. We provide the following dynamic version of their kernel: Our data structure is initialized with an $n$-vertex planar graph $G$ in $O(n \log n)$ amortized time, and, at initialization, outputs a planar graph $K$ with $\mathrm{OPT}(K) = \mathrm{OPT}(G)$ and $|K| = O(\mathrm{OPT}(G))$, where $\mathrm{OPT}(\cdot)$ denotes the size of a minimum dominating set. The graph $G$ can be updated by insertions and deletions of edges and isolated vertices in $O(\log n)$ amortized time per update, under the promise that it remains planar. After each update to $G$, the data structure outputs $O(1)$ updates to $K$, maintaining $\mathrm{OPT}(K) = \mathrm{OPT}(G)$, $|K| = O(\mathrm{OPT}(G))$, and planarity of $K$. Furthermore, we obtain similar dynamic kernelization algorithms for all problems satisfying certain conditions on (topological-)minor-free graph classes. Besides kernelization, this directly implies new dynamic constant-approximation algorithms and improvements to dynamic FPT algorithms for such problems. Our main technical contribution is a dynamic data structure for maintaining an approximately optimal protrusion decomposition of a dynamic topological-minor-free graph. Protrusion decompositions were introduced by Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh, and Thilikos [J. ACM 2016], and have since developed into a part of the core toolbox in kernelization and parameterized algorithms.

Dynamic Meta-Kernelization

TL;DR

This work advances dynamic kernelization by introducing a dynamic protrusion-decomposition framework for H-topological-minor-free graphs, enabling maintenance of approximately optimal treewidth-modulators and efficient protrusion replacements under edge and vertex updates. The core idea is to keep a protrusion decomposition whose root bag tracks a modulator, and to replace large protrusions in one shot via a protrusion replacement automaton, all while updating in amortized O(log |G|) time. The authors prove a meta-theorem: for CMSO-definable problems on CMSO-definable minor- or apex-minor-free classes that are linearly treewidth-bounding with FII, there exist dynamic kernelization data structures producing kernels of size O(OPT(G)) that can be maintained with constant or logarithmic per-update work, and they show direct consequences for dynamic approximation and dynamic FPT algorithms. The framework integrates dynamic treewidth techniques with advanced protrusion-balancing methods, yielding a modular approach that extends static kernelization results to dynamic settings and opens avenues for dynamic preprocessing in sparse graphs. Overall, the paper provides a principled, scalable path to dynamic preprocessing and kernelization, with specific gains for Dominating Set on planar graphs and a broad class of CMSO problems on minor- and topological-minor-free graphs.

Abstract

Kernelization studies polynomial-time preprocessing algorithms. Over the last 20 years, the most celebrated positive results of the field have been linear kernels for classical NP-hard graph problems on sparse graph classes. In this paper, we lift these results to the dynamic setting. As the canonical example, Alber, Fellows, and Niedermeier [J. ACM 2004] gave a linear kernel for dominating set on planar graphs. We provide the following dynamic version of their kernel: Our data structure is initialized with an -vertex planar graph in amortized time, and, at initialization, outputs a planar graph with and , where denotes the size of a minimum dominating set. The graph can be updated by insertions and deletions of edges and isolated vertices in amortized time per update, under the promise that it remains planar. After each update to , the data structure outputs updates to , maintaining , , and planarity of . Furthermore, we obtain similar dynamic kernelization algorithms for all problems satisfying certain conditions on (topological-)minor-free graph classes. Besides kernelization, this directly implies new dynamic constant-approximation algorithms and improvements to dynamic FPT algorithms for such problems. Our main technical contribution is a dynamic data structure for maintaining an approximately optimal protrusion decomposition of a dynamic topological-minor-free graph. Protrusion decompositions were introduced by Bodlaender, Fomin, Lokshtanov, Penninkx, Saurabh, and Thilikos [J. ACM 2016], and have since developed into a part of the core toolbox in kernelization and parameterized algorithms.

Paper Structure

This paper contains 93 sections, 57 theorems, 15 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Linear kernels on sparse graph classes.
  3. Our contribution.
  4. Dynamic meta-kernelization.
  5. Applications outside of kernelization.
  6. Our techniques.
  7. Related work on dynamic kernelization.
  8. Organization of the paper.
  9. Overview
  10. Dynamic protrusion decomposition
  11. The framework and basic operations
  12. Downwards well-linked superbranch decompositions.
  13. From superbranch to protrusion decompositions.
  14. Basic maintenance of the superbranch decomposition.
  15. Controlling the root degree
  16. ...and 78 more sections

Key Result

Theorem 1.1

There is a data structure that is initialized with a planar graph $G$ in $\mathcal{O}(|G| \log |G|)$We denote by $|G| = |V(G)|+|E(G)|$ the total number of vertices and edges of a graph $G$. amortized time, supports updating $G$ via insertions and deletions of edges and isolated vertices in amortized The data structure outputs $K$ at the initialization, and after each update it outputs at most $\ma

Figures (2)

  • Figure 1: Step 2 of the uncrossing: We want to uncross $\mathcal{L}[s]$, where $s$ is a root-child with $\mathsf{int}(\mathcal{L}[s]) \cap \mathsf{bd}(C) \neq \emptyset$. The boundary of $C \setminus \mathcal{L}[s]$ is the boundary of $C$ (blue) without the part that lies in $\mathcal{L}[s]$ (blue, dashed), that is, $\mathsf{bd}(C) \cap \mathsf{int}(\mathcal{L}[s])$, combined with the boundary of $\mathcal{L}[s]$ that lies in $C$ (red, dashed), that is $\mathsf{int}(C) \cap \mathsf{bd}(\mathcal{L}[s])$.
  • Figure 2: Step 3 of the uncrossing: We want to uncross $\mathcal{L}[s]$, where $s$ is a root-child with $\mathsf{int}(\mathcal{L}[s]) \cap \mathsf{bd}(C) = \emptyset$. $\mathcal{A}$ is the union of the internal components of $\mathcal{L}[s]$ that intersect $C$ (which implies $\mathcal{A} \subseteq C$ by \ref{['lem:subsetofB']}), $\mathcal{A}'$ is the union of the remaining components. \ref{['subfig:step3_impossible']} visualizes, why we always have $\mathsf{bd}(\mathcal{A}) \subseteq \mathsf{bd}(\mathcal{A}') = \mathsf{bd}(\mathcal{L}[s])$ or $\mathsf{bd}(\mathcal{A}') \subseteq \mathsf{bd}(\mathcal{A}) = \mathsf{bd}(\mathcal{L}[s])$. In the first case, where $\mathsf{bd}(\mathcal{A}) \subseteq \mathsf{bd}(\mathcal{A}')$ as visualized in \ref{['subfig:step3_exclude']}, we exclude $\mathcal{L}[s]$ from $C$, while in the latter case, where $\mathsf{bd}(\mathcal{A}') \subseteq \mathsf{bd}(\mathcal{A})$ as visualized in \ref{['subfig:step3_include']}, we include $\mathcal{L}[s]$ in $C$.

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.1
  • Theorem 1.1
  • Lemma 3.1: Robertson_Seymour_1995
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 106 more