Preprocessing Complexity for Some Graph Problems Parameterized by Structural Parameters

TL;DR

This paper explores the preprocessing complexity of various fundamental graph problems under different structural parameters, providing a comprehensive landscape of kernelization possibilities.

Contribution

It offers a complete analysis of preprocessing complexity for numerous core problems using various structural graph parameters.

Findings

Polynomial kernels exist for some problems under certain parameters.

Many problems lack polynomial kernels or Turing kernels for key parameters.

The study maps out the complexity landscape for over a dozen problems.

Abstract

Structural graph parameters play an important role in parameterized complexity, including in kernelization. Notably, vertex cover, neighborhood diversity, twin-cover, and modular-width have been studied extensively in the last few years. However, there are many fundamental problems whose preprocessing complexity is not fully understood under these parameters. Indeed, the existence of polynomial kernels or polynomial Turing kernels for famous problems such as Clique, Chromatic Number, and Steiner Tree has only been established for a subset of structural parameters. In this work, we use several techniques to obtain a complete preprocessing complexity landscape for over a dozen of fundamental algorithmic problems.