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Introducing lop-Kernels: A Framework for Kernelization Lower Bounds

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In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain kernelization lower bounds for a certain type of kernels for optimization problems, which we call lop -kernels. Informally, this type of kernel is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. Discrete Appl Math 196:62–71, 2015. https://doi.org/10.1016/j.dam.2014.06.001), and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. We present further applications for Tree Deletion Set and for Maximum Independent Set on \(K_t\)-free graphs. Back to the MMVC problem, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdős–Hajnal property. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless \(\mathsf{NP} \subseteq \mathsf{coNP} / \mathsf{poly}\).

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Notes

  1. This definition was used in the conference version of this paper [4].

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Acknowledgements

We would like to thank Michael Lampis (resp. Magnus Wahlström, Venkatesh Raman) for pointing us to reference [32] (resp. reference [39], references [9, 50]).

Funding

Júlio Araújo: CNPq-Pq 304478/2018-0, CAPES-PrInt 88887.466468/2019-00 and CAPES-STIC-AmSud 88881.569474/2020-01. Victor Campos: FUNCAP - PNE-011200061.01.00/16. Ignasi Sau: DEMOGRAPH (ANR-16-CE40-0028), ESIGMA (ANR-17-CE23-0010), ELIT (ANR-20-CE48-0008-01), and French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027).

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  1. Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Brazil

    Júlio Araújo

  2. LIRMM, CNRS, Université de Montpellier, Montpellier, France

    Marin Bougeret & Ignasi Sau

  3. Departamento de Computação, Universidade Federal do Ceará, Fortaleza, Brazil

    Victor Campos

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  1. Júlio Araújo
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Related version Some of the results of this paper appeared in the preliminary version published in the proceedings of IPEC 2021 [4]. This paper is available at https://arxiv.org/abs/2102.02484

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Araújo, J., Bougeret, M., Campos, V. et al. Introducing lop-Kernels: A Framework for Kernelization Lower Bounds. Algorithmica 84, 3365–3406 (2022). https://doi.org/10.1007/s00453-022-00979-z

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