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Covering Many (Or Few) Edges with k Vertices in Sparse Graphs

Authors Tomohiro Koana , Christian Komusiewicz , André Nichterlein , Frank Sommer

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Author Details

Tomohiro Koana
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Christian Komusiewicz
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany
André Nichterlein
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Frank Sommer
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany

Funding

  • Koana, Tomohiro: Supported by the Deutsche Forschungsgemeinschaft (DFG), project DiPa, NI 369/21.
  • Sommer, Frank: Supported by the Deutsche Forschungsgemeinschaft (DFG), project EAGR, {KO 3669/{6-1}}.

Cite As Get BibTex

Tomohiro Koana, Christian Komusiewicz, André Nichterlein, and Frank Sommer. Covering Many (Or Few) Edges with k Vertices in Sparse Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.STACS.2022.42

Abstract

We study the following two fixed-cardinality optimization problems (a maximization and a minimization variant). For a fixed α between zero and one we are given a graph and two numbers k ∈ ℕ and t ∈ ℚ. The task is to find a vertex subset S of exactly k vertices that has value at least (resp. at most for minimization) t. Here, the value of a vertex set computes as α times the number of edges with exactly one endpoint in S plus 1-α times the number of edges with both endpoints in S. These two problems generalize many prominent graph problems, such as Densest k-Subgraph, Sparsest k-Subgraph, Partial Vertex Cover, and Max (k,n-k)-Cut.
In this work, we complete the picture of their parameterized complexity on several types of sparse graphs that are described by structural parameters. In particular, we provide kernelization algorithms and kernel lower bounds for these problems. A somewhat surprising consequence of our kernelizations is that Partial Vertex Cover and Max (k,n-k)-Cut not only behave in the same way but that the kernels for both problems can be obtained by the same algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized Complexity
  • Kernelization
  • Partial Vertex Cover
  • Densest k-Subgraph
  • Max (k,n-k)-Cut
  • Degeneracy

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