Meta-Kernelization using Well-Structured Modulators

TL;DR

This paper introduces a new class of structural parameters based on well-structured modulators, enabling the derivation of smaller polynomial kernels for graph problems expressible in Monadic Second Order logic.

Contribution

It develops a framework that generalizes modulator size parameters using rank-width and split decompositions, extending kernelization results to broader structural parameters.

Findings

Efficient approximation of well-structured modulators.

Polynomial kernels for MSO-expressible problems using these parameters.

Extension of structural meta-kernelization results.

Abstract

Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters. We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization.