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On 2-Clubs in Graph-Based Data Clustering: Theory and Algorithm Engineering
[article]

2020
*
arXiv
*
pre-print

This naturally leads to the two NP-hard problems 2-

arXiv:2006.14972v1
fatcat:pdshakcombchlcmjmxnwnn4s7e
*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*...##
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On structural parameterizations for the 2-club problem

2015
*
Discrete Applied Mathematics
*

Specifically, we show NP-hardness

doi:10.1016/j.dam.2014.11.026
fatcat:p36xkvzrs5c4jhgqc5rpo5kjpm
*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. ...##
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On Structural Parameterizations for the 2-Club Problem
[chapter]

2013
*
Lecture Notes in Computer Science
*

Specifically, we show NP-hardness

doi:10.1007/978-3-642-35843-2_21
fatcat:aqtmi5ng6beapopjmancpo5slm
*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. ...##
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Finding Disjoint Dense Clubs in an Undirected Graph
[chapter]

2016
*
Lecture Notes in Computer Science
*

Lemma 2 or Lemma 3 then 9:

doi:10.1007/978-3-319-39817-4_27
fatcat:4bd3yxhpmbdrbnxbegfumsjn5q
*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 ...##
###
Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

2015
*
Journal of Graph Algorithms and Applications
*

Given an undirected graph G = (V, E)

doi:10.7155/jgaa.00352
fatcat:5xnvb6pycbf2rjuz35ju7p7tgq
*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 . ...##
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Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs
[chapter]

2012
*
Lecture Notes in Computer Science
*

Given an undirected graph G = (V, E)

doi:10.1007/978-3-642-33293-7_22
fatcat:5hbgpg6wuzbcbetfwpivstuxle
*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 . ...##
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On Structural Parameterizations for the 2-Club Problem
[article]

2013
*
arXiv
*
pre-print

The NP-hard 2-

arXiv:1305.3735v1
fatcat:22pmwuuzfzdjjiqotouhkglqoq
*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. ...##
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Finding large $$k$$ -clubs in undirected graphs

2012
*
Computing
*

In this paper, we show that for fixed k > 1, given a parameter

doi:10.1007/s00607-012-0263-3
fatcat:ew3z5j2p5vc2bdz2q7yjwgxvuu
*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*. ...##
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An Improved Fixed-Parameter Algorithm for 2-Club Cluster Edge Deletion
[article]

2021
*
arXiv
*
pre-print

In the decision version

arXiv:2107.01133v1
fatcat:c77mwvyqdjhd3dn2oiutk4usxq
*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. ...##
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Subexponential Algorithm for d-Cluster Edge Deletion: Exception or Rule?
[chapter]

2013
*
Lecture Notes in Computer Science
*

Our first result shows that assuming the ETH, there is no

doi:10.1007/978-3-642-40313-2_60
fatcat:ipmu2huz4jfmvb6gi6oeo3iyn4
*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*. ...##
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Multivariate Algorithmics for Finding Cohesive Subnetworks

2016
*
Algorithms
*

For each clique relaxation, we give an overview

doi:10.3390/a9010021
fatcat:hqtngvw5pze35blschz3zdvi7i
*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). ...##
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Partitioning Biological Networks into Highly Connected Clusters with Maximum Edge Coverage

2014
*
IEEE/ACM Transactions on Computational Biology & Bioinformatics
*

We show that HIGHLY CONNECTED

doi:10.1109/tcbb.2013.177
pmid:26356014
fatcat:aonwphzqcrhobnuwlrxaigf6my
*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*. ...##
###
Partitioning Biological Networks into Highly Connected Clusters with Maximum Edge Coverage
[chapter]

2013
*
Lecture Notes in Computer Science
*

We show that HIGHLY CONNECTED

doi:10.1007/978-3-642-38036-5_13
fatcat:65xq7e7fazcovlayv4qnqusity
*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*. ...##
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New Races in Parameterized Algorithmics
[chapter]

2012
*
Lecture Notes in Computer Science
*

We discuss several

doi:10.1007/978-3-642-32589-2_2
fatcat:gpfsek5febh5rikaagecihqclq
*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. ...##
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A survey of parameterized algorithms and the complexity of edge modification
[article]

2020
*
arXiv
*
pre-print

The survey provides an overview

arXiv:2001.06867v2
fatcat:6ogysat6n5drpbxsvs63zjmega
*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 ...
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