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Applying Modular Decomposition to Parameterized Cluster Editing Problems

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Abstract

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(4k nm) time by a standard search tree algorithm. We develop an algorithm of time complexity O(4k+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(1.92k+n 3) for this problem was presented by Gramm et al. (Theory Comput. Syst. 38(4), 373–392, 2005, Algorithmica 39(4), 321–347, 2004). In their solution, a problem kernel with O(k 2) vertices is built in O(n 3) time.

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Authors and Affiliations

  1. Instituto de Matemática and Núcleo de Computação Eletrônica, Universidade Federal do Rio de Janeiro, Caixa Postal 2324, 20001-970, Rio de Janeiro, RJ, Brazil

    Fábio Protti & Jayme Luiz Szwarcfiter

  2. COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, 21945-970, Rio de Janeiro, RJ, Brazil

    Maise Dantas da Silva & Jayme Luiz Szwarcfiter

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  1. Fábio Protti
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  2. Maise Dantas da Silva
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  3. Jayme Luiz Szwarcfiter
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Correspondence to Fábio Protti.

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Protti, F., Dantas da Silva, M. & Szwarcfiter, J.L. Applying Modular Decomposition to Parameterized Cluster Editing Problems. Theory Comput Syst 44, 91–104 (2009). https://doi.org/10.1007/s00224-007-9032-7

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  • DOI: https://doi.org/10.1007/s00224-007-9032-7

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