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Applying modular decomposition to parameterized cluster editing problems. (English) Zbl 1179.68111
Summary: A graph $$G$$ is said to be a bicluster graph if $$G$$ is a disjoint union of bicliques (complete bipartite subgraphs), and a cluster graph if $$G$$ is a disjoint union of cliques (complete subgraphs). In this work, we study the parameterized versions of the NP-hard Bicluster Graph Editing and Cluster Graph Editing problems. The former consists of obtaining a bicluster graph by making the minimum number of modifications in the edge set of an input bipartite graph. When at most $$k$$ modifications are allowed (Bicluster$$(k)$$ Graph Editing problem), this problem is FPT, and can be solved in $$O(4^{k } nm)$$ time by a standard search tree algorithm. We develop an algorithm of time complexity $$O(4^{k}+n+m)$$, which uses a strategy based on modular decomposition techniques; we slightly generalize the original problem as the input graph is not necessarily bipartite. The algorithm first builds a problem kernel with $$O(k^{2})$$ vertices in $$O(n+m)$$ time, and then applies a bounded search tree. We also show how this strategy based on modular decomposition leads to a new way of obtaining a problem kernel with $$O(k^{2})$$ vertices for the Cluster$$(k)$$ Graph Editing problem, in $$O(n+m)$$ time. This problem consists of obtaining a cluster graph by modifying at most $$k$$ edges in an input graph. A previous FPT algorithm of time $$O(1.92^{k }+n^{3})$$ for this problem was presented by J. Gramm, J. Guo, F. Hüffner and R. Niedermeier [Theory Comput. Syst. 38, No. 4, 373–392, (2005; Zbl 1084.68117); Algorithmica 39, No. 4, 321–347 (2004; Zbl 1090.68027)]. In their solution, a problem kernel with $$O(k ^{2})$$ vertices is built in $$O(n ^{3})$$ time.

##### MSC:
 68R10 Graph theory (including graph drawing) in computer science 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) 68Q25 Analysis of algorithms and problem complexity
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