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

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

TL;DR

Matching Multicut extends the classic Matching Cut problem by requiring a partition into at least $\ell$ parts where each vertex has at most one neighbor outside its part. The paper develops a multi-faceted complexity picture: NP-hardness on subcubic graphs, a quasi-linear kernel when parameterized by $\ell$, and exact exponential and treewidth-based algorithms. It also advances the enumeration side with PDE kernels for vertex cover and distance-to-co-cluster, and a distance-to-cluster PDE with $2^{O(t^3\log t)}$-delay, together with a kernelization lower bound for the distance-to-cluster parameterization via a Set Packing reduction. The results illuminate the tractability frontier of matching multicuts and open questions about kernel bounds and enumeration on broader graph classes. Overall, the work maps algorithmic strategies across parameterized, exact, and enumerative settings, offering tools and benchmarks for related graph-cut problems.

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 $\ell$ 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 $\ell$ is a parameter, it admits a quasi-linear kernel. We also show an $O(\ell^{\frac{n}{2}})$ time exact exponential algorithm for general graphs and a $2^{O(t \log t)}n^{O(1)}$ time algorithm for graphs of treewidth at most $t$. We then study 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 $\subseteq$ coNP/poly; we highlight that we have no such lower bound for Enum Matching Cut and consider it our main open question.

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

TL;DR

Matching Multicut extends the classic Matching Cut problem by requiring a partition into at least parts where each vertex has at most one neighbor outside its part. The paper develops a multi-faceted complexity picture: NP-hardness on subcubic graphs, a quasi-linear kernel when parameterized by , and exact exponential and treewidth-based algorithms. It also advances the enumeration side with PDE kernels for vertex cover and distance-to-co-cluster, and a distance-to-cluster PDE with -delay, together with a kernelization lower bound for the distance-to-cluster parameterization via a Set Packing reduction. The results illuminate the tractability frontier of matching multicuts and open questions about kernel bounds and enumeration on broader graph classes. Overall, the work maps algorithmic strategies across parameterized, exact, and enumerative settings, offering tools and benchmarks for related graph-cut problems.

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 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 is a parameter, it admits a quasi-linear kernel. We also show an time exact exponential algorithm for general graphs and a time algorithm for graphs of treewidth at most . We then study 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.
Paper Structure (23 sections, 36 theorems, 2 equations, 4 figures)

This paper contains 23 sections, 36 theorems, 2 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. (Sub)Cubic graphs
  4. NP-hardness
  5. Quasi-linear kernel
  6. Exact Exponential Algorithm
  7. FPT Algorithm by Treewidth
  8. Leaf
  9. Introduce node
  10. Join node
  11. Forget node
  12. Matching Multicut Enumeration
  13. Vertex Cover
  14. Distance to Co-cluster
  15. More than three parts
  16. ...and 8 more sections

Key Result

Lemma 1

Let $I = \{H_1, \ldots, H_k\}$ be a set of maximal indivisible pendant subgraphs of $G$. Let $v_1, \ldots, v_k$ be pairwise distinct vertices so that $N(H_i) = \{v_i\}$. If $(G, \ell)$ is a yes-instance for some $\ell > k$, then there exists a matching multicut $\mathcal{P} = \{P_1, \ldots, P_\ell\}

Figures (4)

  • Figure 1: Edge gadget for edge $e_i = uv$. Thick edges are assumed to be in any solution to Matching Multicut.
  • Figure 2: Indivisible pendant subgraph.
  • Figure 3: Subdivided edge.
  • Figure 5: Branching configurations for Matching Multicut.

Theorems & Definitions (64)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 54 more