A 2k Kernel for the Cluster Editing Problem

Chen, Jianer; Meng, Jie

doi:10.1007/978-3-642-14031-0_49

Cited by 36 publications

(48 citation statements)

References 16 publications

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“…Finally, we use a known characterization of graphs that are "locally simple", see Sec. 5. We stress that our result does not hold in general for the Integer-Weighted Cluster Editing problem, as such instances will usually not satisfy the "parity property".…”

Section: Introductionmentioning

confidence: 73%

“…Deleting v x has cost 2, and we branch one last time: Choose another y from K q . Merging v y generates cost 5 2 , deleting v y costs 2. In total, we reach the branching vector (1, 3 2 + 3 2 , 3 2 + 2 + 5 2 , 3 2 + 2 + 2) = (1, 3, 6, 5 1 2 ) with branching number 1.604 < ϕ.…”

Section: Solving Remaining Instancesmentioning

confidence: 99%

“…Using a characterization of graphs that do not contain many conflicts, results in the previously fastest algorithm for Cluster Editing with running time O * (1.76 k ) [3]. A linear kernel with 4k vertices [13] was later improved to 2k vertices [5]. Similarly, a kernel for the integer-weighted variant with O(k 2 ) vertices [1] was improved to 2k vertices [4].…”

Section: Introductionmentioning

confidence: 99%

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A Golden Ratio Parameterized Algorithm for Cluster Editing

Böcker

2011

Lecture Notes in Computer Science

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Abstract. The Cluster Editing problem asks to transform a graph by at most k edge modifications into a disjoint union of cliques. The problem is NP-complete, but several parameterized algorithms are known. We present a novel search tree algorithm for the problem, which improves running time from O(1.76 k + m + n) to O(1.62 k + m + n) for m edges and n vertices. In detail, we can show that we can always branch with branching vector (2, 1) or better, resulting in the golden ratio as the base of the search tree size. Our algorithm uses a well-known transformation to the integer-weighted counterpart of the problem. To achieve our result, we combine three techniques: First, we show that zero-edges in the graph enforce structural features that allow us to branch more efficiently. This is achieved by keeping track of the parity of merged vertices. Second, by repeatedly branching we can isolate vertices, releasing cost. Third, we use a known characterization of graphs with few conflicts. We then show that Integer-Weighted Cluster Editing remains NP-hard for graphs that have a particularly simple structure: namely, a clique minus the edges of a triangle.

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Section: Introductionmentioning

confidence: 73%

Section: Solving Remaining Instancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Golden Ratio Parameterized Algorithm for Cluster Editing

Böcker

2011

Lecture Notes in Computer Science

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“…We finally apply our framework from Section 3 to Cluster Editing, a well-studied edge modification problem in parameterized complexity [7,12,21,28].…”

Section: Cluster Editingmentioning

confidence: 99%

“…Assuming the exponential-time hypothesis, Cluster Editing cannot be solved in 2 o(k) · n O(1) time [21,28]. Cluster Editing admits a problem kernel with at most 2k vertices [12].…”

Section: Problem 61 (Cluster Editing)mentioning

confidence: 99%

Parameterizing Edge Modification Problems Above Lower Bounds

Bevern

Froese

Komusiewicz

2016

Computer Science – Theory and Applications

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We study the parameterized complexity of a variant of the F-free Editing problem: Given a graph G and a natural number k, is it possible to modify at most k edges in G so that the resulting graph contains no induced subgraph isomorphic to F? In our variant, the input additionally contains a vertex-disjoint packing H of induced subgraphs of G, which provides a lower bound h(H) on the number of edge modifications required to transform G into an F-free graph. While earlier works used the number k as parameter or structural parameters of the input graph G, we consider instead the parameter := k − h(H), that is, the number of edge modifications above the lower bound h(H). We develop a framework of generic data reduction rules to show fixed-parameter tractability with respect to for K 3 -Free Editing, Feedback Arc Set in Tournaments, and Cluster Editing when the packing H contains subgraphs with bounded solution size. For K 3 -Free Editing, we also prove NP-hardness in case of edge-disjoint packings of K 3 s and = 0, while for K q -Free Editing and q ≥ 6, NP-hardness for = 0 even holds for vertex-disjoint packings of K q s. In addition, we provide NP-hardness results for F-free Vertex Deletion, were the aim is to delete a minimum number of vertices to make the input graph F-free.

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The Multi-parameterized Cluster Editing Problem

Abu-Khzam

2013

Combinatorial Optimization and Applications

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A 2k Kernel for the Cluster Editing Problem

Cited by 36 publications

References 16 publications

A Golden Ratio Parameterized Algorithm for Cluster Editing

A Golden Ratio Parameterized Algorithm for Cluster Editing

Parameterizing Edge Modification Problems Above Lower Bounds

The Multi-parameterized Cluster Editing Problem

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