A Discharging Method: Improved Kernels for Edge Triangle Packing and Covering

TL;DR

This paper introduces a novel discharging method to improve kernel sizes for edge triangle packing and covering problems, reducing them to 3k vertices, thus advancing parameterized complexity techniques.

Contribution

It presents a new discharging technique that yields smaller kernels for both problems, surpassing previous bounds and offering a tool for future kernel analysis.

Findings

Kernel for Edge Triangle Packing reduced to 3k vertices

Kernel for Edge Triangle Covering reduced to 3k vertices

Discharging method shows potential for analyzing other kernel algorithms

Abstract

\textsc{Edge Triangle Packing} and \textsc{Edge Triangle Covering} are dual problems extensively studied in the field of parameterized complexity. Given a graph $G$ and an integer $k$, \textsc{Edge Triangle Packing} seeks to determine whether there exists a set of at least $k$ edge-disjoint triangles in $G$, while \textsc{Edge Triangle Covering} aims to find out whether there exists a set of at most $k$ edges that intersects all triangles in $G$. Previous research has shown that \textsc{Edge Triangle Packing} has a kernel of $(3+\epsilon)k$ vertices, while \textsc{Edge Triangle Covering} has a kernel of $6k$ vertices. In this paper, we show that the two problems allow kernels of $3k$ vertices, improving all previous results. A significant contribution of our work is the utilization of a novel discharging method for analyzing kernel size, which exhibits potential for analyzing other kernel algorithms.