Crown Structures for Vertex Cover Kernelization

Abu-Khzam, Faisal N.; Fellows, Michael R.; Langston, Michael A.; Suters, W. Henry

doi:10.1007/s00224-007-1328-0

Cited by 113 publications

(112 citation statements)

References 14 publications

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“…This technique was firstly introduced in [1] and [10] and found to be very useful in designing kernelization algorithms for Vertex Cover and related problems [2,9,26].…”

Section: The Decomposition Techniquesmentioning

confidence: 99%

“…A crown decomposition of a graph G is a partition of the vertex set of G into three sets I, C and J such that (1) I is an independent set, (2) there are no edges between I and J, and (3) there is a matching M on the edges between I and C such that all vertices in C are matched.…”

Section: Definition 1 [Crown Decomposition]mentioning

confidence: 99%

“…By this lemma, we can reduce the instance of Vertex Cover by removing I ∪ C of a crown decomposition. There are some methods that find certain crown decompositions of a graph and result in a linear-vertex kernel for Vertex Cover [2]. Figure 1(b).…”

Section: Definition 1 [Crown Decomposition]mentioning

confidence: 99%

“…Then there is no edge between I ′ and J ′ . The partition P satisfies Conditions (2) and (3) in Definition 2 for d-bounded decompositions. Next, we consider Condition We do these iteratively until we find a d-bounded decomposition, where no bad vertex exists.…”

Section: The Algorithm For Decompositionsmentioning

confidence: 99%

“…It has also been proved by Fellows, Guo, Moser and Niedermeier [14] that when d is not bounded, Parameterized Bounded-Degree Vertex Deletion is W[2]-hard. Then unless FPT=W [2], it is impossible to remove d from the size function of any kernel of this problem. These two hardness results also imply that our result is 'tight' in some sense.…”

Section: Introductionmentioning

confidence: 99%

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On a generalization of Nemhauser and Trotter's local optimization theorem

Xiao

2017

Journal of Computer and System Sciences

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“…This technique was firstly introduced in [1] and [10] and found to be very useful in designing kernelization algorithms for Vertex Cover and related problems [2,9,26].…”

Section: The Decomposition Techniquesmentioning

confidence: 99%

Section: Definition 1 [Crown Decomposition]mentioning

confidence: 99%

Section: Definition 1 [Crown Decomposition]mentioning

confidence: 99%

Section: The Algorithm For Decompositionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a generalization of Nemhauser and Trotter's local optimization theorem

Xiao

2017

Journal of Computer and System Sciences

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Critical and Maximum Independent Sets Revisited

Levit

Mândrescu

2019

Mathematical Optimization Theory and Operations Research

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A set S ⊆ V (G) is independent (or stable) if no two vertices from S are adjacent, and by Ind(G) we mean the set of all independent sets of G. A set A ∈ Ind(G) is critical (and we write A ∈ CritIndep(G)) if |A| − |N (A)| = max{|I| − |N (I)| : I ∈ Ind(G)} [54], where N (I) denotes the neighborhood of I. If S ∈ Ind(G) and there is a matching from N (S) into S, then S is a crown [1], and we write S ∈ Crown(G). Let Ψ(G) be the family of all local maximum independent sets of graph G, i.e., S ∈ Ψ(G) if S is a maximum independent set in the subgraph induced by S ∪ N (S) [28]. In this paper we show that CritIndep(G) ⊆ Crown(G) ⊆ Ψ(G) are true for every graph. In addition, we present some classes of graphs where these families coincide and form greedoids or even more general set systems that we call augmentoids.

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Upper and Lower Bounds on Approximating Weighted Mixed Domination

Xiao

2019

Lecture Notes in Computer Science

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A mixed dominating set of a graph G = (V, E) is a mixed set D of vertices and edges, such that for every edge or vertex, if it is not in D, then it is adjacent or incident to at least one vertex or edge in D. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is N P -hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where all vertices are assigned the same nonnegative weight w v and all edges are assigned the same nonnegative weight w e , and the question is to find a mixed dominating set with a minimum total weight. Although the mixed dominating set problem has a simple 2-approximation algorithm, few approximation results for the weighted version are known. The main contributions of this paper include:1. for w e ≥ w v , a 2-approximation algorithm; 2. for w e ≥ 2w v , inapproximability within ratio 1.3606 unless P = N P and within ratio 2 under UGC; 3. for 2w v > w e ≥ w v , inapproximability within ratio 1.1803 unless P = N P and within ratio 1.5 under UGC; 4. for w e < w v , inapproximability within ratio (1 − ) ln |V | unless P = N P for any > 0.

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Crown Structures for Vertex Cover Kernelization

Cited by 113 publications

References 14 publications

On a generalization of Nemhauser and Trotter's local optimization theorem

On a generalization of Nemhauser and Trotter's local optimization theorem

Critical and Maximum Independent Sets Revisited

Upper and Lower Bounds on Approximating Weighted Mixed Domination

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