A new approximate cluster deletion algorithm for diamond-free graphs

The cluster deletion problem (CD) asks for transforming a given graph into a disjoint union of cliques by removing as few edges as possible. CD is among the most studied combinatorial optimization problem and, for general graphs, it is NP-hard. In the present paper, we identify a new polynomially solvable CD subproblem. We specifically propose a two-phase polynomial-time algorithm that optimally solves CD on the class of (butterfly,diamond)-free graphs. For this latter class of graphs, our two-phase algorithm provides optimal solutions even for another clustering variant, namely, cluster editing. Then, we propose a 2-optimal CD algorithm dedicated to the super-class of diamond-free graphs. For this class, we also show that CD, when parameterised by the number of deleted edges, admits a quadratic-size kernel. Finally, we report the results of experiments carried out on numerous diamond-free graphs, showing the effectiveness of the proposed approximate algorithm in terms of solution quality.