Polynomial-Delay Enumeration Kernels for Bounded Degree Cuts in Graphs

Khái niệm cốt lõi

This paper presents polynomial-delay enumeration kernelizations for enumerating all degree-d cuts, all minimal degree-d cuts, and all maximal degree-d cuts in a graph, parameterized by the vertex cover number, neighborhood diversity, and clique partition number of the input graph.

Tóm tắt

The paper considers the Degree-d-Cut problem, which asks whether a given graph has a d-cut, where a d-cut is a bipartition of the vertices such that every vertex in one part has at most d neighbors in the other part. The authors study three enumeration variants of this problem:

1. Enum Deg-d-Cut: Enumerate all the d-cuts of the graph.
2. Enum Min Deg-d-Cut: Enumerate all the minimal d-cuts of the graph.
3. Enum Max Deg-d-Cut: Enumerate all the maximal d-cuts of the graph.

The authors provide the following results:

1. For Enum Min Deg-d-Cut parameterized by the vertex cover number (vc), they give a fully-polynomial enumeration kernel of size O(d^3 * vc^(d+1)).
2. For Enum Deg-d-Cut and Enum Max Deg-d-Cut parameterized by vc, they give polynomial-delay enumeration kernels of size O(d^3 * vc^(d+1)).
3. For all three variants parameterized by the neighborhood diversity (nd), they give polynomial-delay enumeration kernels of size O(d^2 * nd).
4. For all three variants parameterized by the clique partition number (pc), they give bijective enumeration kernelizations of size O(pc^(d+2)).

The authors use structural properties of d-cuts and the marking scheme to establish these results. They also show that these problems do not admit polynomial-delay enumeration kernels of polynomial size when parameterized by treewidth or cliquewidth, unless NP ⊆ coNP/poly.

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Thông tin chi tiết chính được chắt lọc từ

Enumeration Kernels of Polynomial Size for Cuts of Bounded Degree

by Diptapriyo M... lúc arxiv.org 04-30-2024

https://arxiv.org/pdf/2308.01286.pdf

Yêu cầu sâu hơn

Can the techniques used in this paper be extended to other graph cut problems beyond the Degree-d-Cut problem

The techniques used in the paper for the Degree-d-Cut problem can potentially be extended to other graph cut problems. The key lies in identifying the structural parameters and properties of the graph that are relevant to the specific cut problem being considered. By adapting the marking schemes, kernelization algorithms, and proofs to suit the characteristics of the new cut problem, it is possible to develop similar enumeration kernelization results for a broader range of graph cut problems. This approach allows for a systematic study of different cut variants and their enumeration complexities.

Are there any other structural parameters that could lead to polynomial-delay enumeration kernels of polynomial size for these problems

There are several other structural parameters that could potentially lead to polynomial-delay enumeration kernels of polynomial size for graph cut problems. Parameters such as treewidth, modular width, neighborhood diversity, and clique partition number have been shown to be effective in the context of kernelization algorithms. By carefully selecting and analyzing these parameters in relation to the specific graph cut problem at hand, it is possible to develop efficient enumeration kernelization techniques. Additionally, exploring new combinations of parameters and their impact on the enumeration complexity could uncover novel approaches to solving graph cut problems.

What are the implications of these enumeration kernelization results on the complexity of related decision and optimization problems

The enumeration kernelization results obtained in this paper have significant implications for the complexity of related decision and optimization problems. By establishing polynomial-delay enumeration kernels of polynomial size for the Degree-d-Cut problem, it indicates that the enumeration of all minimal, maximal, or general degree-d-cuts can be efficiently achieved within a reasonable time frame. This has a direct impact on the parameterized complexity of these problems, providing insights into the inherent difficulty of enumerating solutions based on specific structural parameters of the input graph. Furthermore, the techniques and methodologies developed in this paper can be applied to other graph problems, potentially leading to advancements in the field of parameterized enumeration and kernelization.