Improved Kernels for Edge Modification Problems

TL;DR

This paper develops improved kernelization algorithms for various edge modification problems, reducing the problem size significantly and providing more efficient solutions for graph editing tasks.

Contribution

The paper introduces new, smaller kernels for multiple edge modification problems, improving upon previous kernel sizes and complexity bounds.

Findings

A 2k-vertex kernel for cluster edge deletion.

A 3k^2-vertex kernel for trivially perfect completion.

A 5k^{1.5}-vertex kernel for split and pseudo-split problems.

Abstract

In an edge modification problem, we are asked to modify at most $k$ edges to a given graph to make the graph satisfy a certain property. Depending on the operations allowed, we have the completion problems and the edge deletion problems. A great amount of efforts have been devoted to understanding the kernelization complexity of these problems. We revisit several well-studied edge modification problems, and develop improved kernels for them: \begin{itemize} \item a $2 k$-vertex kernel for the cluster edge deletion problem, \item a $3 k^2$-vertex kernel for the trivially perfect completion problem, \item a $5 k^{1.5}$-vertex kernel for the split completion problem and the split edge deletion problem, and \item a $5 k^{1.5}$-vertex kernel for the pseudo-split completion problem and the pseudo-split edge deletion problem. \end{itemize} Moreover, our kernels for split completion and pseudo-split completion have only $O(k^{2.5})$ edges. Our results also include a $2 k$-vertex kernel for the strong triadic closure problem, which is related to cluster edge deletion.