A New Framework for Kernelization Lower Bounds: The Case of Maximum Minimal Vertex Cover

Authors Júlio Araújo , Marin Bougeret , Victor Campos , Ignasi Sau

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Author Details

Júlio Araújo
  • Departament of Mathematics, Federal University of Ceará, Fortaleza, Brazil
Marin Bougeret
  • LIRMM, Université de Montpellier, France
Victor Campos
  • Departament of Computer Science, Federal University of Ceará, Fortaleza, Brazil
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France


We would like to thank Michael Lampis (resp. Magnus Wahlström, Venkatesh Raman) for pointing us to reference [Louis Dublois et al., 2020] (resp. reference [Archontia C. Giannopoulou et al., 2016], references [Arindam Biswas et al., 2020; Stefan Kratsch et al., 2014]).

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Júlio Araújo, Marin Bougeret, Victor Campos, and Ignasi Sau. A New Framework for Kernelization Lower Bounds: The Case of Maximum Minimal Vertex Cover. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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 lower bounds on the degrees of a certain type of polynomial kernels for vertex-optimization problems, which we call {lop-kernels}. Informally, this type of kernels 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. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. On the positive side, 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 in order to find an appropriate decomposition. 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 NP ⊆ coNP / poly. This indicates that parameters smaller than the solution size are unlike to yield polynomial kernels for MMVC.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Maximum minimal vertex cover
  • parameterized complexity
  • polynomial kernel
  • kernelization lower bound
  • Erdős-Hajnal property
  • induced subgraphs


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