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A Shortcut to (Sun)Flowers: Kernels in Logarithmic Space or Linear Time

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Mathematical Foundations of Computer Science 2015 (MFCS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9235))

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 kernel 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 ).

Supported by the Emmy Noether-program of the German Research Foundation (DFG), research project PREMOD (KR 4286/1).

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Notes

  1. 1.

    Proofs of statements marked with \(\bigstar \) are in the full paper [4].

  2. 2.

    It is well known that finding a k-sunflower is \(\mathsf {NP}\)-hard in general. Similarly, finding a k-flower is \(\mathsf {coNP}\)-hard. For self-contained proofs see the full paper [4]. Both proofs do not apply when the size of the set family exceeds the bounds in the (sun)flower lemma.

  3. 3.

    Note that we would run into the same obstacle if we use sunflowers instead of flowers; finding out whether there exists a \((k+1)\)-sunflower with core \(C \subseteq F\) for a specific set F is \(\mathsf {NP}\)-hard as we show in the full paper [4].

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Authors and Affiliations

  1. University of Bonn, Bonn, Germany

    Stefan Fafianie & Stefan Kratsch

Authors
  1. Stefan Fafianie
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  2. Stefan Kratsch
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Correspondence to Stefan Fafianie .

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Editors and Affiliations

  1. University of Rome "Tor Vergata", Rome, Italy

    Giuseppe F. Italiano

  2. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  3. University of Edinburgh, Edinburgh, United Kingdom

    Donald T. Sannella

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Fafianie, S., Kratsch, S. (2015). A Shortcut to (Sun)Flowers: Kernels in Logarithmic Space or Linear Time. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_25

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  • DOI: https://doi.org/10.1007/978-3-662-48054-0_25

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