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Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

Authors Eva-Maria C. Hols , Stefan Kratsch , Astrid Pieterse

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Author Details

Eva-Maria C. Hols
  • Department of Computer Science, Humboldt-Universität zu Berlin, Germany
Stefan Kratsch
  • Department of Computer Science, Humboldt-Universität zu Berlin, Germany
Astrid Pieterse
  • Department of Computer Science, Humboldt-Universität zu Berlin, Germany

Funding

  • Hols, Eva-Maria C.: Supported by DFG Emmy Noether-grant (KR 4286/1).
  • Pieterse, Astrid: Supported by NWO Gravitation grant "Networks".

Cite AsGet BibTex

Eva-Maria C. Hols, Stefan Kratsch, and Astrid Pieterse. Elimination Distances, Blocking Sets, and Kernels for Vertex Cover. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.STACS.2020.36

Abstract

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets. A blocking set is a set of vertices such that no optimal vertex cover contains all vertices in the blocking set, and the study of minimal blocking sets played implicit and explicit roles in many existing results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class ?, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class ?, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes ?, including the class ?_{LP} of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or ?_{LP} graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Vertex Cover
  • kernelization
  • blocking sets
  • elimination distance
  • structural parameters

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References

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