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DOI: 10.4230/LIPIcs.SWAT.2024.6

Destroying Densest Subgraphs Is Hard

Authors: Cristina Bazgan, André Nichterlein, and Sofia Vazquez Alferez

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

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.

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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)

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@InProceedings{bazgan_et_al:LIPIcs.SWAT.2024.6,
  author =	{Bazgan, Cristina and Nichterlein, Andr\'{e} and Vazquez Alferez, Sofia},
  title =	{{Destroying Densest Subgraphs Is Hard}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.6},
  URN =		{urn:nbn:de:0030-drops-200461},
  doi =		{10.4230/LIPIcs.SWAT.2024.6},
  annote =	{Keywords: Graph modification problems, NP-hardness, fixed-parameter tractability, W-hardness, special graph classes}
}

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Destroying Densest Subgraphs Is Hard

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