Destroying Densest Subgraphs is Hard
<|reference_start|>Destroying Densest Subgraphs is Hard: 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 $\tau_\rho$, are there $k$ edges ($k$ vertices) whose removal from $G$ results in a graph where the densest subgraph has density at most $\tau_\rho$? 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.<|reference_end|>
@article{bazgan2024destroying,
    title={Destroying Densest Subgraphs is Hard},
    author={Cristina Bazgan, Andr'e Nichterlein, Sofia Vazquez Alferez},
    journal={arXiv preprint arXiv:2404.08599},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.08599},
    primaryClass={cs.DS}
}