[Paper Review] Kernelization Using Structural Parameters on Sparse Graph Classes

Gajarsk\'y, Jakub, Hlin\v{e}n\'y, Petr+6 more|arXiv (Cornell University)|Feb 27, 2013

Advanced Graph Theory Research30 references33 citations

TL;DR

This paper establishes meta-theorems for linear and almost-linear kernelization on sparse graph classes by introducing a novel parameter: the size of a modulator to constant treedepth. It proves that any graph problem with finite integer index (FII) on constant treedepth graphs admits a linear kernel on bounded expansion graphs and an almost-linear kernel on nowhere dense graphs when parameterized by this structural modulator, overcoming limitations of previous parameters like treewidth modulators.

ABSTRACT

Meta-theorems for polynomial (linear) kernels have been the subject of intensive research in parameterized complexity. Heretofore, meta-theorems for linear kernels exist on graphs of bounded genus, $H$-minor-free graphs, and $H$-topological-minor-free graphs. To the best of our knowledge, no meta-theorems for polynomial kernels are known for any larger sparse graph classes; e.g., for classes of bounded expansion or for nowhere dense ones. In this paper we prove such meta-theorems for the two latter cases. More specifically, we show that graph problems that have finite integer index (FII) have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. While our parameter may seem rather strong, we argue that a linear kernelization result on graphs of bounded expansion with a weaker parameter (than treedepth modulator) would fail to include some of the problems covered by our framework. Moreover, we only require the problems to have FII on graphs of constant treedepth. This allows us to prove linear kernels for problems such as Longest Path/Cycle, Exact $s,t$-Path, Treewidth, and Pathwidth, which do not have FII on general graphs (and the first two not even on bounded treewidth graphs).

Motivation & Objective

• To address the gap in meta-theorems for polynomial kernels on larger sparse graph classes such as bounded expansion and nowhere dense graphs.
• To overcome the limitations of previous kernelization parameters—particularly treewidth modulators—which fail to support linear kernels for natural problems like Longest Path and Treewidth.
• To identify a structural parameter that both increases under edge subdivisions and enables linear kernelization for FII problems on sparse graph classes.
• To unify and generalize existing kernelization meta-theorems by focusing on FII on constant treedepth graphs as a foundational property.

Proposed method

• Introduce the treedepth modulator as a structural parameter that increases under edge subdivisions, unlike treewidth modulators.
• Prove that problems with finite integer index (FII) on graphs of constant treedepth admit linear kernels on bounded expansion graphs via a refined protrusion replacement technique.
• Leverage the stability of FII under graph operations such as disjoint union and edge contraction, using bounded treewidth and pathwidth as base cases.
• Apply a recursive decomposition strategy based on tree decompositions and minimal separators to control the width of subgraphs after modulator removal.
• Use the concept of t-boundaried graphs and equivalence relations (≃pw,t, ≡pw,t) to bound the number of distinct graph types that need to be considered in kernelization.
• Establish that treewidth and pathwidth have FII on graphs of bounded treewidth, enabling the kernelization framework to be applied to these problems.

Experimental results

Research questions

• RQ1 Can meta-theorems for linear kernels be extended to larger sparse graph classes such as bounded expansion and nowhere dense graphs?
• RQ2 Is the treedepth modulator a viable parameter for achieving linear kernels in sparse graph classes where treewidth modulators fail?
• RQ3 Do problems like Longest Path and Treewidth, which lack FII on general or bounded treewidth graphs, still admit linear kernels when parameterized by a treedepth modulator?
• RQ4 Can the FII property on constant treedepth graphs serve as a unifying condition for kernelization meta-theorems across sparse graph classes?
• RQ5 What is the relationship between FII on constant treedepth graphs and the existence of almost-linear kernels on nowhere dense graph classes?

Key findings

• All graph problems with finite integer index (FII) on graphs of constant treedepth admit linear kernels on bounded expansion graphs when parameterized by the size of a modulator to constant treedepth.
• For nowhere dense graph classes, the same framework yields almost-linear kernels, improving upon previous results that required stronger structural assumptions.
• The treedepth modulator parameter is essential: a linear kernelization result using a treewidth modulator on bounded expansion graphs would fail to include natural problems like Feedback Vertex Set and Treewidth t-Vertex Deletion.
• Pathwidth and treewidth problems have FII on graphs of bounded treewidth, which enables the application of the kernelization framework to these fundamental problems.
• The framework generalizes prior meta-theorems by unifying the underlying condition (FII on constant treedepth) across different sparse graph classes.
• The results demonstrate that FII on constant treedepth graphs is a sufficient and structurally sound condition for kernelization in sparse graph classes, even when problems lack FII on general or bounded treewidth graphs.