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On 2-Clubs in Graph-Based Data Clustering: Theory and Algorithm Engineering
[article]
2020
arXiv
pre-print
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This naturally leads to the two NP-hard problems 2-Club Cluster Editing (the allowed editing operations are edge insertion and edge deletion) and 2-Club Cluster Vertex Deletion (the allowed editing operations ...
Then focusing on 2-Club Cluster Vertex Deletion, which is easily seen to be fixed-parameter tractable, we show that under standard complexity-theoretic assumptions it does not have a polynomial-size problem ...
An edge set F ⊆ V 2 such that G[E F ] is an s-club cluster graph is called an s-club editing set and a vertex set S ⊆ V such that G[V \ S] is an s-club cluster graph is called an s-club vertex deletion ...
arXiv:2006.14972v1
fatcat:pdshakcombchlcmjmxnwnn4s7e
On structural parameterizations for the 2-club problem
2015
Discrete Applied Mathematics
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Specifically, we show NP-hardness of 2-Club on graphs that become bipartite by deleting one vertex, on graphs that can be covered by three cliques, and on graphs with domination number two and diameter ...
Finally, we show that 2-Club is fixed-parameter tractable with respect to "distance to co-cluster graphs" and "distance to cluster graphs". ...
For the relevant notions of parameterized complexity we refer to [9, 16] . For the parameters distance to cluster/co-cluster graph we assume that a deletion set is provided as an additional input. ...
doi:10.1016/j.dam.2014.11.026
fatcat:p36xkvzrs5c4jhgqc5rpo5kjpm
On Structural Parameterizations for the 2-Club Problem
[chapter]
2013
Lecture Notes in Computer Science
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Specifically, we show NP-hardness of 2-Club on graphs that become bipartite by deleting one vertex, on graphs that can be covered by three cliques, and on graphs with domination number two and diameter ...
Finally, we show that 2-Club is fixed-parameter tractable with respect to "distance to co-cluster graphs" and "distance to cluster graphs". ...
For the relevant notions of parameterized complexity we refer to [9, 16] . For the parameters distance to cluster/co-cluster graph we assume that a deletion set is provided as an additional input. ...
doi:10.1007/978-3-642-35843-2_21
fatcat:aqtmi5ng6beapopjmancpo5slm
Finding Disjoint Dense Clubs in an Undirected Graph
[chapter]
2016
Lecture Notes in Computer Science
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Lemma 2 or Lemma 3 then 9: delete one vertex of this edge, form G and call Branching(G , s, t, k − 1) 10: delete the other vertex of this edge and form G and call Branching(G , s, t, k − 1) 11: else 12 ...
The paper [19] extends the previous parameterized complexity study for 2-club and provides polynomial-size kernels for 2-club parameterized by "cluster editing set size of G" and "size of a cluster editing ...
doi:10.1007/978-3-319-39817-4_27
fatcat:4bd3yxhpmbdrbnxbegfumsjn5q
Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs
2015
Journal of Graph Algorithms and Applications
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Given an undirected graph G = (V, E) and an integer ≥ 1, the NPhard 2-Club problem asks for a vertex set S ⊆ V of size at least such that the subgraph induced by S has diameter at most two. ...
On the positive side, we give polynomial-size problem kernels for the parameters feedback edge set size of G and size of a cluster editing set of G and present a direct combinatorial algorithm for the ...
case s-Club Cluster Vertex Deletion is equivalent to s-Club parameterized by the dual parameter k . ...
doi:10.7155/jgaa.00352
fatcat:5xnvb6pycbf2rjuz35ju7p7tgq
Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs
[chapter]
2012
Lecture Notes in Computer Science
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Given an undirected graph G = (V, E) and an integer ≥ 1, the NPhard 2-Club problem asks for a vertex set S ⊆ V of size at least such that the subgraph induced by S has diameter at most two. ...
On the positive side, we give polynomial-size problem kernels for the parameters feedback edge set size of G and size of a cluster editing set of G and present a direct combinatorial algorithm for the ...
case s-Club Cluster Vertex Deletion is equivalent to s-Club parameterized by the dual parameter k . ...
doi:10.1007/978-3-642-33293-7_22
fatcat:5hbgpg6wuzbcbetfwpivstuxle
On Structural Parameterizations for the 2-Club Problem
[article]
2013
arXiv
pre-print
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The NP-hard 2-Club problem is, given an undirected graph G=(V,E) and l∈ N, to decide whether there is a vertex set S⊆ V of size at least l such that the induced subgraph G[S] has diameter at most two. ...
We make progress towards a systematic classification of the complexity of 2-Club with respect to a hierarchy of prominent structural graph parameters. ...
A set of edge insertions and deletions is a cluster editing set if it transforms G into a cluster graph. A set of edges is a feedback edge set if its deletion results in a graph without any cycle. ...
arXiv:1305.3735v1
fatcat:22pmwuuzfzdjjiqotouhkglqoq
Finding large $$k$$ -clubs in undirected graphs
2012
Computing
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In this paper, we show that for fixed k > 1, given a parameter s and a graph G, to determine whether G has a k-club of size greater than or equal to s is fixed-parameter tractable. ...
It supports vertex deletion. Given a graph G = (V, E), deleting a vertex u from G, the data structure maintains all vertices at distance at most k from v in G[V \ {u}]. ...
[4] gave a branch-and-bound algorithm for solving the maximum k-club problem without analyzing the time complexity. In this section, we analyze the worst-case time complexity of this algorithm. ...
doi:10.1007/s00607-012-0263-3
fatcat:ew3z5j2p5vc2bdz2q7yjwgxvuu
An Improved Fixed-Parameter Algorithm for 2-Club Cluster Edge Deletion
[article]
2021
arXiv
pre-print
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In the decision version of the parametrized 2-Club Cluster Edge Deletion problem, an undirected graph G is given along with an integer k≥ 0 as parameter, and the question is whether G can be transformed ...
into a disjoint union of 2-clubs by deleting at most k edges. ...
Many variants of editing a graph into a disjoint union of 2-clubs have been studied, such as 2-CLUB CLUSTER VERTEX DELETION, 2-CLUB CLUSTER EDGE DELETION, and 2-CLUB CLUSTER EDITING. ...
arXiv:2107.01133v1
fatcat:c77mwvyqdjhd3dn2oiutk4usxq
Subexponential Algorithm for d-Cluster Edge Deletion: Exception or Rule?
[chapter]
2013
Lecture Notes in Computer Science
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Our first result shows that assuming the ETH, there is no algorithm solving the s-Club Cluster Edge Deletion problem in time 2 o(k) n O(1) . ...
We show, further, that even the problem of deleting edges to obtain a graph with d s-clubs cannot be solved in time 2 o(k) n O(1) for any fixed s, d ≥ 2. ...
The complexity of Cluster Edge Deletion and Cluster Editing is well-understood. The problems are NP-complete and admit constant-factor approximation algorithms. ...
doi:10.1007/978-3-642-40313-2_60
fatcat:ipmu2huz4jfmvb6gi6oeo3iyn4
Multivariate Algorithmics for Finding Cohesive Subnetworks
2016
Algorithms
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For each clique relaxation, we give an overview of basic model properties and of the complexity of the problem of finding large cohesive subgraphs under this model. ...
Since this problem is usually NP-hard, we focus on combinatorial fixed-parameter algorithms exploiting typical structural properties of input networks. ...
Acknowledgments: Christian Komusiewicz was supported by the DFG project "Multivariate Algorithmics for Graph and String Problems in Bioinformatics" (KO 3669/4-1). ...
doi:10.3390/a9010021
fatcat:hqtngvw5pze35blschz3zdvi7i
Partitioning Biological Networks into Highly Connected Clusters with Maximum Edge Coverage
2014
IEEE/ACM Transactions on Computational Biology & Bioinformatics
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We show that HIGHLY CONNECTED DELETION is NP-hard and provide a fixed-parameter algorithm and a kernelization. ...
A popular clustering algorithm for biological networks which was proposed by Hartuv and Shamir [IPL 2000] identifies nonoverlapping highly connected components. ...
We also thank Andrea Kappes (Karlsruhe Institute of Technology) for pointing out a flaw in a previous version of the column generation algorithm. ...
doi:10.1109/tcbb.2013.177
pmid:26356014
fatcat:aonwphzqcrhobnuwlrxaigf6my
Partitioning Biological Networks into Highly Connected Clusters with Maximum Edge Coverage
[chapter]
2013
Lecture Notes in Computer Science
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We show that HIGHLY CONNECTED DELETION is NP-hard and provide a fixed-parameter algorithm and a kernelization. ...
A popular clustering algorithm for biological networks which was proposed by Hartuv and Shamir [IPL 2000] identifies nonoverlapping highly connected components. ...
We also thank Andrea Kappes (Karlsruhe Institute of Technology) for pointing out a flaw in a previous version of the column generation algorithm. ...
doi:10.1007/978-3-642-38036-5_13
fatcat:65xq7e7fazcovlayv4qnqusity
New Races in Parameterized Algorithmics
[chapter]
2012
Lecture Notes in Computer Science
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We discuss several of these aspects and particularly focus on the search for "stronger parameterizations" in developing fixed-parameter algorithms. ...
Herein, the attention usually focuses on improving the running time factor exponential in the considered parameter, and, in case of kernelization algorithms, to improve the bound on the kernel size. ...
We thank André Nichterlein, Manuel Sorge, and Mathias Weller for their comments which have improved this article. ...
doi:10.1007/978-3-642-32589-2_2
fatcat:gpfsek5febh5rikaagecihqclq
A survey of parameterized algorithms and the complexity of edge modification
[article]
2020
arXiv
pre-print
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The survey provides an overview of the developing area of parameterized algorithms for graph modification problems. ...
We concentrate on edge modification problems, where the task is to change a small number of adjacencies in a graph in order to satisfy some required property. ...
We thank Marcin Pilipczuk, William Lochet, and Dekel Tsur for helpful comments.
References ...
arXiv:2001.06867v2
fatcat:6ogysat6n5drpbxsvs63zjmega
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