Efficient Kernelization Algorithms for Modifying Graphs to Have at Most Two Distinct Eigenvalues

Основные понятия

This paper presents efficient kernelization algorithms for modifying a given graph to have at most two distinct eigenvalues by vertex deletion, edge editing, edge deletion, and edge addition.

Аннотация

The paper studies the problem of modifying a given graph G = (V, E) such that the resulting graph has at most two distinct eigenvalues. This is equivalent to transforming G into a collection of equal-sized cliques. The authors consider four different graph modification operations: vertex deletion (2-Eigenvalue Vertex Deletion, 2-EVD), edge editing (2-Eigenvalue Edge Editing, 2-EEE), edge deletion (2-Eigenvalue Edge Deletion, 2-EED), and edge addition (2-Eigenvalue Edge Addition, 2-EEA).
For the 2-EVD problem, the authors provide a kernel of size O(k^3), where k is the solution size (the number of vertices to be deleted). For the 2-EEE and 2-EED problems, they provide kernels of size O(k^2). Finally, for the 2-EEA problem, they provide a linear kernel of size 6k.
The key steps in the kernelization algorithms include:

Identifying a maximal set of vertex-disjoint induced P3s (paths on 3 vertices) in the graph, as a graph is a cluster graph if and only if it does not contain an induced P3.
Applying various reduction rules to bound the number of cliques in the graph based on their sizes and the vertices' neighborhoods.
Exploiting the fact that in a yes-instance, the vertices in the graph can be partitioned into two sets: those with degree equal to the average degree, and those with degree not equal to the average degree.

The authors also discuss the connections between their results and previous work on related problems, such as Balanced Cluster Completion and s-Club Cluster Vertex Deletion.

Статистика

The paper does not contain any explicit numerical data or statistics. The focus is on developing efficient kernelization algorithms for the considered graph modification problems.

Цитаты

There are no direct quotes from the paper that are particularly striking or support the key logics.

Ключевые выводы из

Kernelization Algorithms for the Eigenvalue Deletion Problems

by Ajinkya Gaik... в arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10023.pdf

Дополнительные вопросы

Can the kernelization algorithms be further improved to achieve better upper bounds on the kernel sizes

The kernelization algorithms presented in the paper provide upper bounds on the kernel sizes for the 2-Eigenvalue Vertex Deletion (2-EVD), 2-Eigenvalue Edge Editing (2-EEE), and 2-Eigenvalue Edge Deletion (2-EED) problems. While the bounds achieved are already quite efficient, there is always room for improvement. One potential avenue for enhancing the kernelization algorithms could be to explore more sophisticated reduction rules that can further streamline the graph modification process. By identifying additional patterns or structures in the graphs that allow for more aggressive reductions, it may be possible to reduce the kernel sizes even further. Additionally, optimizing the existing reduction rules to be more efficient in their application could lead to smaller kernels. Experimenting with different combinations of reduction rules and exploring their interactions could also potentially yield better upper bounds on the kernel sizes.

Are there any connections between the considered graph modification problems and other well-studied graph problems, such as graph coloring or graph partitioning, that could be exploited to develop more efficient algorithms

There are indeed connections between the considered graph modification problems, such as 2-EVD, 2-EEE, and 2-EED, and other well-studied graph problems like graph coloring and graph partitioning that could be leveraged to develop more efficient algorithms. For example, techniques used in graph coloring, such as identifying cliques or independent sets, could be applied to the graph modification problems to simplify the structure of the graph. Graph partitioning algorithms, which aim to divide a graph into subgraphs with certain properties, could also be adapted to assist in the modification tasks by breaking down the graph into more manageable components. By drawing parallels between these different types of graph problems and utilizing common algorithmic strategies, it may be possible to devise more effective and optimized approaches for solving the graph modification problems considered in the paper.

How do the techniques developed in this paper compare to the approaches used for solving similar graph modification problems in the literature, and are there any insights that could be transferred to other related problems

The techniques developed in this paper for solving the 2-Eigenvalue Deletion Problems demonstrate a deep understanding of graph structures and eigenvalues, leading to efficient kernelization algorithms. These techniques showcase the importance of identifying specific graph properties, such as cliques and equal-sized components, to simplify the graph modification tasks. The approach of reducing the graph to a collection of disjoint cliques through targeted vertex and edge deletions or additions is a powerful strategy that can be applied to various graph modification problems.
Comparing these techniques to approaches used for similar graph modification problems in the literature, we see a focus on leveraging the inherent structure of the graph to achieve efficient solutions. By building on the insights gained from studying eigenvalues and cluster graphs, the developed techniques offer a systematic and parameterized approach to graph modification. These insights could potentially be transferred to other related problems in graph theory, particularly those involving graph decomposition, spectral analysis, and structural modifications. By adapting the principles and reduction strategies employed in this paper to different graph problems, researchers may uncover new algorithmic tools and methodologies for addressing a wide range of graph modification challenges.