Correlation Clustering Algorithm for Dynamic Complete Signed Graphs: An Index-based Approach

In this paper, we reduce the complexity of approximating the correlation clustering problem from $O(m\times\left( 2+ \alpha (G) \right)+n)$ to $O(m+n)$ for any given value of $\varepsilon$ for a complete signed graph with $n$ vertices and $m$ positive edges where $\alpha(G)$ is the arboricity of the graph. Our approach gives the same output as the original algorithm and makes it possible to implement the algorithm in a full dynamic setting where edge sign flipping and vertex addition/removal are allowed. Constructing this index costs $O(m)$ memory and $O(m\times\alpha(G))$ time. We also studied the structural properties of the non-agreement measure used in the approximation algorithm. The theoretical results are accompanied by a full set of experiments concerning seven real-world graphs. These results shows superiority of our index-based algorithm to the non-index one by a decrease of %34 in time on average.

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.

A note on belief structures and S-approximation spaces

We study relations between evidence theory and S-approximation spaces. Both theories have their roots in the analysis of Dempster's multivalued mappings and lower and upper probabilities and have close relations to rough sets. We show that an S-approximation space, satisfying a monotonicity condition, can induce a natural belief structure which is a fundamental block in evidence theory. We also demonstrate that one can induce a natural belief structure on one set, given a belief structure on another set if those sets are related by a partial monotone S-approximation space.