(Meta) Kernelization

TL;DR

This paper presents two meta-theorems that unify and extend kernelization results for parameterized problems on graphs of bounded genus, focusing on problems expressible in Counting Monadic Second Order Logic and those with finite integer index.

Contribution

It introduces two general meta-theorems providing polynomial and linear kernels for broad classes of problems on bounded genus graphs, extending prior results.

Findings

Problems in Counting Monadic Second Order Logic with coverability admit polynomial kernels.

Problems with finite integer index and weaker coverability admit linear kernels.

Unified and extended kernelization results for planar graph problems.

Abstract

In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two meta-theorems on kernelzation. The first theorem says that all problems expressible in Counting Monadic Second Order Logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.