Computational Complexity of Destroying Densest Subgraphs

Основні поняття

Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion problems are computationally hard, with polynomial-time algorithms only for specific graph classes and target densities.

Анотація

The paper analyzes the computational complexity of two problems called Bounded-Density Edge Deletion and Bounded-Density Vertex Deletion. In these problems, given a graph G, a budget k, and a target density τρ, the goal is to find a set of k edges (vertices) whose removal from G results in a graph where the densest subgraph has density at most τρ.
The key findings are:

Polynomial-time algorithms:

For Bounded-Density Edge Deletion, the problem can be solved in polynomial time for specific target densities below 1 (e.g., 0, 1/2, 2/3) or when the input is a tree or clique.
For Bounded-Density Vertex Deletion, the problem can be solved in linear time for trees.

NP-hardness results:

Bounded-Density Edge Deletion is NP-complete for target densities above or below 1, even on claw-free cubic planar, planar bipartite, and split graphs.
Bounded-Density Vertex Deletion is NP-complete for all target densities in the range [0, n^(1-1/c)] for any constant c.

Parameterized complexity:

Bounded-Density Edge Deletion is W[1]-hard with respect to the combined parameter of solution size k and feedback edge number, but fixed-parameter tractable with respect to the vertex cover number.
Bounded-Density Vertex Deletion is W[2]-hard with respect to k, but fixed-parameter tractable with respect to the vertex cover number.

The results demonstrate that the problems are computationally challenging in general, with polynomial-time algorithms only for specific graph classes and target densities.

Статистика

None.

Цитати

None.

Ключові висновки, отримані з

Destroying Densest Subgraphs is Hard

by Cris... о arxiv.org 04-15-2024

https://arxiv.org/pdf/2404.08599.pdf

Глибші Запити

What other graph parameters or structural properties could be exploited to obtain efficient algorithms for these problems

One graph parameter that could be exploited to obtain efficient algorithms for these problems is the treewidth of the graph. Graphs with low treewidth have a more restricted structure, making them amenable to dynamic programming techniques. By leveraging the treewidth of the graph, algorithms can be designed to efficiently solve problems like Bounded-Density Edge Deletion and Vertex Deletion. Additionally, exploiting properties like planarity, degree constraints, or specific graph classes can also lead to algorithmic optimizations. For example, in the context of planar graphs, algorithms can take advantage of planar embedding and planar graph properties to devise more efficient solutions.

How do the hardness results change if the target density τρ is allowed to be a variable input, rather than a fixed constant

If the target density τρ is allowed to be a variable input rather than a fixed constant, the hardness results for the problems may change. Allowing τρ to be a variable input introduces additional complexity to the problems as the density constraint becomes more flexible. This flexibility can lead to a wider range of possible solutions and potentially increase the computational complexity of finding an optimal solution. The problems may become more challenging to solve as the target density varies, requiring more sophisticated algorithms to handle the variable nature of τρ.

Are there any real-world applications or practical scenarios where these problems arise, and how could the insights from this work be leveraged to address them

These problems have applications in various real-world scenarios such as network analysis, social network modeling, and optimization in various domains. For example, in social network analysis, identifying dense subgraphs can help in understanding community structures or identifying influential nodes. In network optimization, removing edges or vertices to achieve a desired density can be useful in resource allocation or network restructuring. The insights from this work can be leveraged in developing algorithms for community detection, anomaly detection, or network optimization tasks in practical scenarios. By applying the findings from this research, practitioners can enhance their network analysis capabilities and make informed decisions based on the densest subgraph properties.