Competitive Online Clique Clustering

Clique clustering is the problem of partitioning a graph into cliques so that some objective function is optimized. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The objective here is to maintain a clustering the never deviates too far in the objective function compared to the optimal solution. We give a constant competitive upper bound for online clique clustering, where the objective function is to maximize the number of edges inside the clusters. We also give almost matching upper and lower bounds on the competitive ratio for online clique clustering, where we want to minimize the number of edges between clusters. In addition, we prove that the greedy method only gives linear competitive ratio for these problems.