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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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