Robust Correlation Clustering Problem with Locally Bounded Disagreements

Min-max disagreements are an important generalization of the correlation clustering problem (CorCP). It can be defined as follows. Given a marked complete graph $G=(V, E)$ , each edge in the graph is marked by a positive label “+” or a negative label “−” based on the similarity of the connected vertices. The goal is to find a clustering $\mathcal{C}$ of vertices $V$ , so as to minimize the number of disagreements at the vertex with the most disagreements. Here, the disagreements are the positive cut edges and the negative non-cut edges produced by clustering $\mathcal{C}$ . This paper considers two robust min-max disagreements: min-max disagreements with outliers and min-max disagreements with penalties. Given parameter $\delta\in(0,1/14)$ , we first provide a threshold-based iterative clustering algorithm based on LP-rounding technique, which is a $(1/\delta, 7/(1-14\delta))$ -bi-criteria approximation algorithm for both the min-max disagreements with outliers and the min-max disagreements with outliers on one-sided complete bipartite graphs. Next, we verify that the above algorithm can achieve an approximation ratio of 21 for min-max disagreements with penalties when we set parameter $\delta=1/21$ .