Optimal Algorithms and Lower Bounds for Bounded Degree Vertex Deletion and Defective Coloring

Core Concepts

The core message of this paper is that the standard dynamic programming algorithms for Bounded Degree Vertex Deletion and Defective Coloring are essentially optimal when parameterized by treewidth, pathwidth, tree-depth, and vertex cover.

Abstract

The paper revisits two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring, where the input is a graph G and a target degree Δ, and the goal is to edit or partition the graph so that the maximum degree becomes bounded by Δ. Both problems are known to be parameterized intractable for the most well-known structural parameters, such as treewidth.
The authors present a more fine-grained picture of the complexity of both problems:

For treewidth and pathwidth, they show that the standard dynamic programming algorithms are essentially optimal. Specifically, they prove that under the SETH, no algorithm can solve Bounded Degree Vertex Deletion and Defective Coloring in time (Δ+2-ε)^pw and (χd(Δ+1)-ε)^pw respectively, where pw is the pathwidth of the input graph, and χd is the number of available colors for Defective Coloring.

For tree-depth, the authors show that under the ETH, neither problem can be solved in time no(td), where td is the tree-depth of the input graph. This implies that the treewidth-based algorithm remains (qualitatively) optimal even in this more restricted case.

For vertex cover, the authors prove that the best known algorithms, which have a super-exponential dependence on the vertex cover, are optimal under the ETH. This is shown by a new application of the technique of d-detecting families.

Overall, the results paint a clear picture of the structurally parameterized complexity of these two problems, indicating that the standard dynamic programming algorithms are optimal in a multitude of restricted cases.

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Key Insights Distilled From

Structural Parameterizations for Two Bounded Degree Problems Revisited

by Michael Lamp... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2304.14724.pdf

Deeper Inquiries

Are there other problems that exhibit a similar behavior, where the standard dynamic programming algorithm is provably optimal for various structural parameters

In the realm of parameterized complexity, there are indeed other problems that exhibit a similar behavior where the standard dynamic programming algorithm is proven to be optimal for various structural parameters. Problems like Vertex Cover, k-Coloring, and Dominating Set fall into this category. For instance, in Vertex Cover, the standard dynamic programming algorithm is known to be optimal when parameterized by tree width. Similarly, for k-Coloring, the standard DP algorithm is optimal when parameterized by treewidth or pathwidth. Dominating Set also showcases this behavior, where the standard DP algorithm is the best-known approach for certain structural parameters. These problems demonstrate that the standard DP algorithm can be highly effective and optimal for specific structural parameterizations.

Can the techniques used in this paper be applied to obtain tight lower bounds for other problems that involve bounding the degree or partitioning a graph into low-degree components

The techniques utilized in the paper can be applied to obtain tight lower bounds for other problems that involve bounding the degree or partitioning a graph into low-degree components. By leveraging concepts like treewidth, pathwidth, and tree-depth, researchers can analyze the complexity of various graph problems and determine the optimal algorithms for different structural parameters. For problems related to graph partitioning, such as Edge Deletion to a Planar Graph or Vertex Partitioning, similar techniques can be employed to establish lower bounds and understand the algorithmic limits for these scenarios. Additionally, the use of d-detecting families, as demonstrated in the paper, can be extended to other problems to provide tight lower bounds based on specific structural parameters.

What are the implications of these results for the design of practical algorithms for Bounded Degree Vertex Deletion and Defective Coloring in real-world applications

The results presented in the paper have significant implications for the design of practical algorithms for Bounded Degree Vertex Deletion and Defective Coloring in real-world applications. Understanding the structural parameterizations and the complexity landscape of these problems allows algorithm designers to tailor their approaches effectively. By knowing that the standard dynamic programming algorithm is optimal for certain parameters, developers can focus on refining and optimizing this approach rather than searching for alternative methods. This insight can lead to the development of more efficient algorithms for solving Bounded Degree Vertex Deletion and Defective Coloring in practical applications, ensuring better performance and scalability in real-world graph-related tasks.