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Enumeration Kernels of Polynomial Size for Cuts of Bounded Degree

Christian Komusiewicz, Diptapriyo Majumdar

TL;DR

The paper addresses the enumeration of all $d$-cuts in graphs under a parameterized complexity framework. It applies polynomial-delay and fully-polynomial enumeration kernelization to the three variants ENUM $d$-Cut, ENUM MIN-$d$-Cut, and ENUM MAX-$d$-Cut, across three structural parameters: ${\textsf{vc}}$, ${\textsf{nd}}$, and ${\textsf{pc}}$. It provides concrete kernel-size and delay bounds: for ${\textsf{vc}}$, a fully-polynomial kernel for ENUM MIN-$d$-Cut; polynomial-delay kernels for ENUM $d$-Cut and ENUM MAX-$d$-Cut; and bijective kernels for all three under ${\textsf{pc}}$, with analogous results for ${\textsf{nd}}$. These results extend the study of enumeration kernelization to the broader $d$-Cut setting and relate to the Matching Cut case when $d=1$, offering compact representations of all solutions under natural graph parameters.

Abstract

Enumeration kernelization was first proposed by Creignou et al. [TOCS 2017] and was later refined by Golovach et al. [JCSS 2022] into two different variants: fully-polynomial enumeration kernelization and polynomial-delay enumeration kernelization. In this paper, we consider the d-CUT problem from the perspective of (polynomial-delay) enumeration kenrelization. Given an undirected graph G = (V, E), a cut F = (A, B) is a d-cut of G if every $u \in A$ has at most d neighbors in B and every $v \in B$ has at most d neighbors in A. Checking the existence of a d-cut in a graph is a well-known NP-hard problem and is well-studied in parameterized complexity [Algorithmica 2021, IWOCA 2021]. This problem also generalizes a well-studied problem MATCHING CUT (set d = 1) that has been a central problem in the literature of polynomial-delay enumeration kernelization. In this paper, we study three different enumeration variants of this problem, ENUM d-CUT, ENUM MIN-d-CUT and ENUM MAX-d-CUT that intends to enumerate all the d-cuts, all the minimal d-cuts and all the maximal d-cuts respectively. We consider various structural parameters of the input, e.g. vertex cover number, neighborhood diversity, and clique partition number. When vertex cover number and neighborhood diversity are considered as parameters, we provide polynomial-delay enumeration kernelizations of polynomial size for ENUM d-CUT and ENUM MAX-d-CUT and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-d-CUT. When clique partition number is considered as the parameter, we provide bijective enumeration kernels for each of these three problems.

Enumeration Kernels of Polynomial Size for Cuts of Bounded Degree

TL;DR

The paper addresses the enumeration of all -cuts in graphs under a parameterized complexity framework. It applies polynomial-delay and fully-polynomial enumeration kernelization to the three variants ENUM -Cut, ENUM MIN--Cut, and ENUM MAX--Cut, across three structural parameters: , , and . It provides concrete kernel-size and delay bounds: for , a fully-polynomial kernel for ENUM MIN--Cut; polynomial-delay kernels for ENUM -Cut and ENUM MAX--Cut; and bijective kernels for all three under , with analogous results for . These results extend the study of enumeration kernelization to the broader -Cut setting and relate to the Matching Cut case when , offering compact representations of all solutions under natural graph parameters.

Abstract

Enumeration kernelization was first proposed by Creignou et al. [TOCS 2017] and was later refined by Golovach et al. [JCSS 2022] into two different variants: fully-polynomial enumeration kernelization and polynomial-delay enumeration kernelization. In this paper, we consider the d-CUT problem from the perspective of (polynomial-delay) enumeration kenrelization. Given an undirected graph G = (V, E), a cut F = (A, B) is a d-cut of G if every has at most d neighbors in B and every has at most d neighbors in A. Checking the existence of a d-cut in a graph is a well-known NP-hard problem and is well-studied in parameterized complexity [Algorithmica 2021, IWOCA 2021]. This problem also generalizes a well-studied problem MATCHING CUT (set d = 1) that has been a central problem in the literature of polynomial-delay enumeration kernelization. In this paper, we study three different enumeration variants of this problem, ENUM d-CUT, ENUM MIN-d-CUT and ENUM MAX-d-CUT that intends to enumerate all the d-cuts, all the minimal d-cuts and all the maximal d-cuts respectively. We consider various structural parameters of the input, e.g. vertex cover number, neighborhood diversity, and clique partition number. When vertex cover number and neighborhood diversity are considered as parameters, we provide polynomial-delay enumeration kernelizations of polynomial size for ENUM d-CUT and ENUM MAX-d-CUT and fully-polynomial enumeration kernels of polynomial size for ENUM MIN-d-CUT. When clique partition number is considered as the parameter, we provide bijective enumeration kernels for each of these three problems.
Paper Structure (9 sections, 6 theorems, 1 figure, 1 table)

This paper contains 9 sections, 6 theorems, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our Contributions and Motivations Behind the Choice of Parameters.
  3. Further Related Work.
  4. Preliminaries
  5. Sets, Numbers, and Graph Theory
  6. Some Fundamental Properties of $d$-Cuts
  7. Parameterized Complexity, Kernelization, and Parameterized Enumeration
  8. Parameterized Complexity and Kernelization.
  9. Parameterized Enumeration and Enumeration Kernelization.

Key Result

Theorem 1

For every fixed $d \geq 1$, Enum Min-$d$-Cut parameterized by ${\textsf{vc}}$ admits a fully-polynomial enumeration kernel with $\mathcal{O}(d {\textsf{vc}}^2)$ vertices.

Figures (1)

  • Figure 1: An overview of parameter hierarchy. A green box shows the existence of (polynomial-delay) enumeration kernelizations of polynomial size. An orange box indicates non-existence of enumeration kernelizations of polynomial size unless ${\rm NP \subseteq coNP/poly}$. A pink box indicates the NP-hardness for a constant parameter value. A yellow box indicates an open status of the existence of (polynomial-delay) enumeration kernelizations of polynomial size. The results of this paper are marked with a $\star$.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Definition 9