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Destroying Densest Subgraphs Is Hard

Authors Cristina Bazgan , André Nichterlein , Sofia Vazquez Alferez

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Author Details

Cristina Bazgan
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, Paris, France
André Nichterlein
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Sofia Vazquez Alferez
  • Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, Paris, France

Acknowledgements

We thank anonymous reviewers of SWAT 2024 for their detailed comments improving the presentation.

Cite AsGet BibTex

Cristina Bazgan, André Nichterlein, and Sofia Vazquez Alferez. Destroying Densest Subgraphs Is Hard. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SWAT.2024.6

Abstract

We analyze the computational complexity of the following computational problems called Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion: Given a graph G, a budget k and a target density τ_ρ, are there k edges (k vertices) whose removal from G results in a graph where the densest subgraph has density at most τ_ρ? Here, the density of a graph is the number of its edges divided by the number of its vertices. We prove that both problems are polynomial-time solvable on trees and cliques but are NP-complete on planar bipartite graphs and split graphs. From a parameterized point of view, we show that both problems are fixed-parameter tractable with respect to the vertex cover number but W[1]-hard with respect to the solution size. Furthermore, we prove that Bounded-Density Edge Deletion is W[1]-hard with respect to the feedback edge number, demonstrating that the problem remains hard on very sparse graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Graph modification problems
  • NP-hardness
  • fixed-parameter tractability
  • W-hardness
  • special graph classes

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