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

Aleksander Figiel, Anne-Sophie Himmel, André Nichterlein, Rolf Niedermeier
2020 arXiv   pre-print
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

Sepp Hartung, Christian Komusiewicz, André Nichterlein, Ondřej Suchý
2015 Discrete Applied Mathematics  
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]

Sepp Hartung, Christian Komusiewicz, André Nichterlein
2013 Lecture Notes in Computer Science  
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]

Peng Zou, Hui Li, Chunlin Xin, Wencheng Wang, Binhai Zhu
2016 Lecture Notes in Computer Science  
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

Sepp Hartung, Christian Komusiewicz, André Nichterlein
2015 Journal of Graph Algorithms and Applications  
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]

Sepp Hartung, Christian Komusiewicz, André Nichterlein
2012 Lecture Notes in Computer Science  
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]

Sepp Hartung, Christian Komusiewicz, André Nichterlein, Ondrej Suchý
2013 arXiv   pre-print
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

Maw-Shang Chang, Ling-Ju Hung, Chih-Ren Lin, Ping-Chen Su
2012 Computing  
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]

Faisal N. Abu-Khzam, Norma Makarem, Maryam Shehab
2021 arXiv   pre-print
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]

Neeldhara Misra, Fahad Panolan, Saket Saurabh
2013 Lecture Notes in Computer Science  
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

Christian Komusiewicz
2016 Algorithms  
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

Falk Huffner, Christian Komusiewicz, Adrian Liebtrau, Rolf Niedermeier
2014 IEEE/ACM Transactions on Computational Biology & Bioinformatics  
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]

Falk Hüffner, Christian Komusiewicz, Adrian Liebtrau, Rolf Niedermeier
2013 Lecture Notes in Computer Science  
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]

Christian Komusiewicz, Rolf Niedermeier
2012 Lecture Notes in Computer Science  
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]

Christophe Crespelle, Pål Grønås Drange, Fedor V. Fomin, Petr A. Golovach
2020 arXiv   pre-print
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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