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Acta Informatica
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Fixed-parameter tractability of graph modification problems for hereditary properties
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FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
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Discrete Applied Mathematics - Discrete mathematics & data mining (DM & DM)
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Theory of Computing Systems
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Correlation clustering in general weighted graphs
Theoretical Computer Science - Approximation and online algorithms
Invitation to data reduction and problem kernelization
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Journal of the ACM (JACM)
Going weighted: Parameterized algorithms for cluster editing
Theoretical Computer Science
Applying modular decomposition to parameterized bicluster editing
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
The lost continent of polynomial time: preprocessing and kernelization
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Efficient parameterized preprocessing for cluster editing
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
A more effective linear kernelization for Cluster Editing
ESCAPE'07 Proceedings of the First international conference on Combinatorics, Algorithms, Probabilistic and Experimental Methodologies
Fixed-parameter tractability of multicut parameterized by the size of the cutset
Proceedings of the forty-third annual ACM symposium on Theory of computing
CGGA'10 Proceedings of the 9th international conference on Computational Geometry, Graphs and Applications
Finding small separators in linear time via treewidth reduction
ACM Transactions on Algorithms (TALG)
Hi-index | 5.23 |
The Correlation Clustering problem, also known as the Cluster Editing problem, seeks to edit a given graph by adding and deleting edges to obtain a collection of vertex-disjoint cliques, such that the editing cost is minimized. The Edge Clique Partitioning problem seeks to partition the edges of a given graph into edge-disjoint cliques, such that the number of cliques is minimized. Both problems are known to be NP-hard, and they have been previously studied with respect to approximation and fixed-parameter tractability. In this paper we study these two problems in a more general setting that we term fuzzy graphs, where the input graphs may have missing information, meaning that whether or not there is an edge between some pairs of vertices of the input graph can be undecided. For fuzzy graphs the Correlation Clustering and Edge Clique Partitioning problems have previously been studied only with respect to approximation. Here we give parameterized algorithms based on kernelization for both problems. We prove that the Correlation Clustering problem is fixed-parameter tractable on fuzzy graphs when parameterized by (k,r), where k is the editing cost and r is the minimum number of vertices required to cover the undecided edges. In particular we show that it has a polynomial-time reduction to a problem kernel on O(k^2+r) vertices. We provide an analogous result for the Edge Clique Partitioning problem on fuzzy graphs. Using (k,r) as parameters, where k bounds the size of the partition, and r is the minimum number of vertices required to cover the undecided edges, we describe a polynomial-time kernelization to a problem kernel on O(k^4@?3^r) vertices. This implies fixed-parameter tractability for this parameterization. Furthermore we also show that parameterizing only by the number of cliques k, is not enough to obtain fixed-parameter tractability. The problem remains, in fact, NP-hard for each fixed k2.