eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-05-31
6:1
6:17
10.4230/LIPIcs.SWAT.2024.6
article
Destroying Densest Subgraphs Is Hard
Bazgan, Cristina
1
https://orcid.org/0000-0002-5460-6222
Nichterlein, André
2
https://orcid.org/0000-0001-7451-9401
Vazquez Alferez, Sofia
1
https://orcid.org/0000-0002-1541-8683
Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243, LAMSADE, Paris, France
Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol294-swat2024/LIPIcs.SWAT.2024.6/LIPIcs.SWAT.2024.6.pdf
Graph modification problems
NP-hardness
fixed-parameter tractability
W-hardness
special graph classes