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An Improved Kernel and Parameterized Algorithm for Almost Induced Matching

Yuxi Liu, Mingyu Xiao

TL;DR

The paper addresses the Almost Induced Matching problem parameterized by the deletion budget $k$. It introduces an AIM crown decomposition to obtain a $6k$-vertex kernel and develops a branch-and-bound algorithm with a worst-case runtime of $O^*(1.6765^k)$ using polynomial space. The core contributions are the linear-vertex kernel and the refined branching strategy that surpass previous bounds of $7k$ and $O^*(1.7485^k)$. These advances improve practical feasibility for AIM and sharpen the theoretical understanding of kernelization and FPT algorithms in this domain.

Abstract

An induced subgraph is called an induced matching if each vertex is a degree-1 vertex in the subgraph. The \textsc{Almost Induced Matching} problem asks whether we can delete at most $k$ vertices from the input graph such that the remaining graph is an induced matching. This paper studies parameterized algorithms for this problem by taking the size $k$ of the deletion set as the parameter. First, we prove a $6k$-vertex kernel for this problem, improving the previous result of $7k$. Second, we give an $O^*(1.6765^k)$-time and polynomial-space algorithm, improving the previous running-time bound of $O^*(1.7485^k)$.

An Improved Kernel and Parameterized Algorithm for Almost Induced Matching

TL;DR

The paper addresses the Almost Induced Matching problem parameterized by the deletion budget . It introduces an AIM crown decomposition to obtain a -vertex kernel and develops a branch-and-bound algorithm with a worst-case runtime of using polynomial space. The core contributions are the linear-vertex kernel and the refined branching strategy that surpass previous bounds of and . These advances improve practical feasibility for AIM and sharpen the theoretical understanding of kernelization and FPT algorithms in this domain.

Abstract

An induced subgraph is called an induced matching if each vertex is a degree-1 vertex in the subgraph. The \textsc{Almost Induced Matching} problem asks whether we can delete at most vertices from the input graph such that the remaining graph is an induced matching. This paper studies parameterized algorithms for this problem by taking the size of the deletion set as the parameter. First, we prove a -vertex kernel for this problem, improving the previous result of . Second, we give an -time and polynomial-space algorithm, improving the previous running-time bound of .
Paper Structure (7 sections, 14 theorems, 32 equations, 6 figures, 2 tables)

This paper contains 7 sections, 14 theorems, 32 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kernelization
  4. A parameterized algorithm
  5. Branching rules
  6. The algorithm
  7. Conclusion

Key Result

lemma thmcounterlemma

If graph $G = (V, E)$ has an independent set $I \subseteq V$ with $|I| > |N(I)|$, then a VC crown decomposition $(C, H, R)$ with $\emptyset \neq C \subseteq I$ and $H \subseteq N(I)$ can be found in linear time.

Figures (6)

  • Figure 1: An AIM crown decomposition $(C, H, R)$, where the three bold edges form an injective matching from $H$ to $C$.
  • Figure 2: Sets $S^*$ and $S'$ in the proof of Lemma \ref{['AIM-crown-lemma']}.
  • Figure 3: Algorithm ${\tt Reduce}(G,k)$
  • Figure 4: Step 5. Case 1.
  • Figure 5: Step 5. Case 2: Vertices in the deletion set are denoted by black vertices.
  • ...and 1 more figures

Theorems & Definitions (23)

  • definition thmcounterdefinition
  • lemma thmcounterlemma: chor2004linear
  • lemma thmcounterlemma: xiao2020parameterized
  • lemma thmcounterlemma: xiao2020parameterized
  • lemma thmcounterlemma: xiao2020parameterized
  • definition thmcounterdefinition: AIM crown decomposition
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • ...and 13 more