On the Approximability of Maximum and Minimum Edge Clique Partition Problems

Dessmark, Anders; Jansson, Jesper; Lingas, Andrzej; Lundell, Eva-Marta; Persson, Mia

doi:10.1142/s0129054107004656

Cited by 24 publications

(25 citation statements)

References 16 publications

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“…More precisely, Maximum Edge Clique Partition can be seen as the 1-familymatching problem in general graphs. As observed by [12], in an optimal solution of Maximum Edge Clique Partition, the maximum size of the clusters is not far from the size of a maximum clique of the input graph, which can be used in order to show that this problem cannot be approximated within n 1− for any unless P = N P [31]. This approach could be used in Inria order to show that the 2-family-matching problem is not approximable within any ratio by using Maximum Edge Biclique as starting problem, in which the aim is to find a complete bipartite graph with the maximum number of edges in a bipartite graph.…”

Section: Main Contributionssupporting

confidence: 56%

Comparing Two Clusterings Using Matchings between Clusters of Clusters

Cazals¹,

Mazauric²,

Tetley³

et al. 2019

ACM J. Exp. Algorithmics

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Clustering is a fundamental problem in data science, yet, the variety of clustering methods and their sensitivity to parameters make clustering hard. To analyze the stability of a given clustering algorithm while varying its parameters, and to compare clusters yielded by different algorithms, several comparison schemes based on matchings, information theory and various indices (Rand, Jaccard) have been developed. We go beyond these by providing a novel class of methods computing meta-clusters within each clustering-a meta-cluster is a group of clusters, together with a matching between these. Let the intersection graph of two clusterings be the edge-weighted bipartite graph in which the nodes represent the clusters, the edges represent the non empty intersection between two clusters, and the weight of an edge is the number of common items. We introduce the so-called D-family-matching problem on intersection graphs, with D the upper-bound on the diameter of the graph induced by the clusters of any meta-cluster. First we prove NP-completeness results and unbounded approximation ratio of simple strategies. Second, we design exact polynomial time dynamic programming algorithms for some classes of graphs (in particular trees). Then, we prove spanning-tree based efficient algorithms for general graphs. Our experiments illustrate the role of D as a scale parameter providing information on the relationship between clusters within a clustering and in-between two clusterings. They also show the advantages of our built-in mapping over classical cluster comparison measures such as the variation of information (VI). Comparer deux clusterings en utilisant des clusters de clustersRésumé :Le clustering est une tâche essentielle en analyse de données, mais la variété des méthodes disponibles rend celle-ci ardue. Diverses stratégies ont été proposées pour analyser la stabilité d'un clustering en fonction des paramètres de l'algorithme l'ayant généré, ou bien comparer des clusterings produits par des algorithmes différents. Nous allons au delà de celles-ci, en proposant une nouvelle classe de méthodes formant des groupes de clusters (meta-clusters) dans chaque clustering, et établissant une correspondance entre ceux-ci.Plus spécifiquement, définissons le graphe intersection de deux clusterings comme le graphe biparti dont les sommets sont les clusters, chaque arête étant pondérée par le nombre de points communs à deux clusters. Nous définissons le D-family-matching problème à partir du graphe intersection, D étant une borne supérieure sur le diamètre du graphe induit par les clusters des metaclusters. Dans un premier temps, nous établissons des résultats de difficulté et d'inaproximabilité. Dans un second temps, nous développons des algorithmes de programmation dynamique pour certaines classes de graphes (arbres en particulier). Enfin, nous concevons des algorithmes efficaces, basés sur des arbres couvrants, pour des graphes généraux.Nos résultats expérimentaux illustrent le rôle de D comme un paramètre d'échelle fournissant de l'i...

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Section: Main Contributionssupporting

confidence: 56%

Comparing Two Clusterings Using Matchings between Clusters of Clusters

Cazals¹,

Mazauric²,

Tetley³

et al. 2019

ACM J. Exp. Algorithmics

View full text Add to dashboard Cite

show abstract

“…However, we feel that since the lower bound adversarial arguments also allow this computation, our measure is fairer to the strategy. In addition, the computational problems required to be solved efficiently for our strategy are indeed efficiently solvable for large classes of graphs, such as chordal graphs, line graphs and circular-arc graphs [7].…”

Section: Resultsmentioning

confidence: 99%

“…These measures give rise to the maximum and minimum edge clique partition problems (Max-ECP and Min-ECP for short) respectively; the computitional complexity and approximability of the aforementioned problems have attracted significant attention recently [7,9,11], and they have numerous applications within the areas of gene expression profiling and DNA clone classification [1,3,8,11].…”

Section: Introductionmentioning

confidence: 99%

Competitive Online Clique Clustering

Fabijan

Nilsson

Persson

2013

Lecture Notes in Computer Science

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Abstract. Clique clustering is the problem of partitioning a graph into cliques so that some objective function is optimized. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The objective here is to maintain a clustering the never deviates too far in the objective function compared to the optimal solution. We give a constant competitive upper bound for online clique clustering, where the objective function is to maximize the number of edges inside the clusters. We also give almost matching upper and lower bounds on the competitive ratio for online clique clustering, where we want to minimize the number of edges between clusters. In addition, we prove that the greedy method only gives linear competitive ratio for these problems.

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“…By exhaustively analyzing the behavior of Strategy OCC in phase 1, taking into account that γ ≈ 3.303 > 3, we can establish that R 1 = 10. We will then bound the remaining ratios using a refined version of recurrence (12).…”

Section: Absolute Competitive Ratiomentioning

confidence: 99%

Online Clique Clustering

et al. 2019

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Clique clustering is the problem of partitioning the vertices of a graph into disjoint clusters, where each cluster forms a clique in the graph, while optimizing some objective function. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The goal is to maintain a clustering with an objective value close to the optimal solution. For the variant where we want to maximize the number of edges in the clusters, we propose an online strategy based on the doubling technique. It has an asymptotic competitive ratio at most 15.646 and an absolute competitive ratio at most 22.641. We also show that no deterministic strategy can have an asymptotic competitive ratio better than 6. For the variant where we want to minimize the number of edges between clusters, we show that the deterministic competitive ratio of the problem is n − ω(1), where n is the number of vertices in the graph.

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On the Approximability of Maximum and Minimum Edge Clique Partition Problems

Cited by 24 publications

References 16 publications

Comparing Two Clusterings Using Matchings between Clusters of Clusters

Comparing Two Clusterings Using Matchings between Clusters of Clusters

Competitive Online Clique Clustering

Online Clique Clustering

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