an:05590696
Zbl 1179.68111
Protti, F??bio; Dantas da Silva, Maise; Szwarcfiter, Jayme Luiz
Applying modular decomposition to parameterized cluster editing problems
EN
Theory Comput. Syst. 44, No. 1, 91-104 (2009).
00244592
2009
j
68R10 68Q17 68Q25
NP-complete problems; fixed-parameter tractability; edge modification problems; cluster graphs; bicluster graphs
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 \textit{J. Gramm, J. Guo, F. H??ffner} and \textit{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.
Zbl 1084.68117; Zbl 1090.68027