A shortcut to (sun)flowers: Kernels in logarithmic space or linear time

TL;DR

This paper explores the possibility of kernelization algorithms operating in logarithmic space and presents new logspace kernelizations for several problems, along with simplified and improved linear-time kernelizations.

Contribution

It introduces the first logspace kernelizations for multiple problems and simplifies existing linear-time kernelizations, improving efficiency and understanding.

Findings

Logspace kernelizations for d-Hitting Set, d-Set Packing, and Edge Dominating Set.

Simplified linear-time kernelization for d-Hitting Set.

New linear-time kernel for d-Set Packing.

Abstract

We investigate whether kernelization results can be obtained if we restrict kernelization algorithms to run in logarithmic space. This restriction for kernelization is motivated by the question of what results are attainable for preprocessing via simple and/or local reduction rules. We find kernelizations for d-Hitting Set(k), d-Set Packing(k), Edge Dominating Set(k) and a number of hitting and packing problems in graphs, each running in logspace. Additionally, we return to the question of linear-time kernelization. For d-Hitting Set(k) a linear-time kernelization was given by van Bevern [Algorithmica (2014)]. We give a simpler procedure and save a large constant factor in the size bound. Furthermore, we show that we can obtain a linear-time kernel for d-Set Packing(k) as well.