Approximation Algorithms for Clique Clustering

A clique clustering of a graph is a partitioning of its vertices into disjoint cliques. The quality of a clique clustering is measured by the total number of edges in its cliques. We consider the online variant of the clique clustering problem, where the vertices of the input graph arrive one at a time. At each step, the newly arrived vertex forms a singleton clique, and the algorithm can merge any existing cliques in its partitioning into larger cliques, but splitting cliques is not allowed. We give an online algorithm with competitive ratio 15.645 and we prove a lower bound of 6 on the competitive ratio, improving the previous respective bounds of 31 and 2.