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Matching (Multi)Cut: Algorithms, Complexity, and Enumeration

Authors Guilherme C. M. Gomes , Emanuel Juliano , Gabriel Martins , Vinicius F. dos Santos

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Author Details

Guilherme C. M. Gomes
  • Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
  • CNRS/LIRMM, Montpellier, France
Emanuel Juliano
  • Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Gabriel Martins
  • Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Vinicius F. dos Santos
  • Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

Funding

  • C. M. Gomes, Guilherme: Partially funded by the European Union, project PACKENUM, grant number 101109317. Views and opinions expressed are however those of the author only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them.
  • F. dos Santos, Vinicius: Partially funded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), grants 312069/2021-9, 406036/2021-7 and 404479/2023-5 by Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG), grant APQ-01707-21.

Cite As Get BibTex

Guilherme C. M. Gomes, Emanuel Juliano, Gabriel Martins, and Vinicius F. dos Santos. Matching (Multi)Cut: Algorithms, Complexity, and Enumeration. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.25

Abstract

A matching cut of a graph is a partition of its vertex set in two such that no vertex has more than one neighbor across the cut. The Matching Cut problem asks if a graph has a matching cut. This problem, and its generalization d-cut, has drawn considerable attention of the algorithms and complexity community in the last decade, becoming a canonical example for parameterized enumeration algorithms and kernelization. In this paper, we introduce and study a generalization of Matching Cut, which we have named Matching Multicut: can we partition the vertex set of a graph in at least 𝓁 parts such that no vertex has more than one neighbor outside its part? We investigate this question in several settings. We start by showing that, contrary to Matching Cut, it is NP-hard on cubic graphs but that, when 𝓁 is a parameter, it admits a quasi-linear kernel. We also show an 𝒪(𝓁^{n/2}) time exact exponential algorithm for general graphs and a 2^{𝒪(tlog t)}n^{𝒪(1)} time algorithm for graphs of treewidth at most t. We then turn our attention to parameterized enumeration aspects of matching multicuts. First, we generalize the quadratic kernel of Golovach et. al for Enum Matching Cut parameterized by vertex cover, then use it to design a quadratic kernel for Enum Matching (Multi)cut parameterized by vertex-deletion distance to co-cluster. Our final contributions are on the vertex-deletion distance to cluster parameterization, where we show an FPT-delay algorithm for Enum Matching Multicut but that no polynomial kernel exists unless NP ⊆ coNP/poly; we highlight that we have no such lower bound for Enum Matching Cut and consider it our main open question.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Matching Cut
  • Matching Multicut
  • Enumeration
  • Parameterized Complexity
  • Exact exponential algorithms

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