Online Correlation Clustering for Dynamic Complete Signed Graphs

In the correlation clustering problem for complete signed graphs, the input is a complete signed graph with edges weighted as $+1$ (denote recommendation to put this pair in the same cluster) or $-1$ (recommending to put this pair of vertices in separate clusters) and the target is to cluster the set of vertices such that the number of disagreements with these recommendations is minimized. In this paper, we consider the problem of correlation clustering for dynamic complete signed graphs where (1) a vertex can be added or deleted, and (2) the sign of an edge can be flipped. In the proposed online scheme, the offline approximation algorithm in [CALM+21] for correlation clustering is used. Up to the author's knowledge, this is the first online algorithm for dynamic graphs which allows a full set of graph editing operations. The proposed approach is rigorously analyzed and compared with a baseline method, which runs the original offline algorithm on each time step. Our results show that the dynamic operations have local effects on the neighboring vertices and we employ this locality to reduce the dependency of the running time in the Baseline to the summation of the degree of all vertices in $G_t$, the graph after applying the graph edit operation at time step $t$, to the summation of the degree of the changing vertices (e.g. two endpoints of an edge) and the number of clusters in the previous time step. Moreover, the required working memory is reduced to the square of the summation of the degree of the modified edge endpoints rather than the total number of vertices in the graph.