Simple Algorithms for Bad Triangle Transversals with Applications to Correlation Clustering

Correlation clustering is a classic approach for summarizing signed graphs, where the goal is to cluster the graph while minimizing positive inter-cluster edges plus negative intra-cluster edges. On complete signed graphs, correlation clustering is closely related to the bad triangle traversal (BTT) problem of finding the smallest number of edges that need to be removed such that the remaining graph does not have a bad triangle. Here, a bad triangle is a triangle with exactly one negative edge. A known result states that a feasible bad triangle cover $F$ on a complete signed graph can be transformed into a correlation clustering with at most $2|F|$ mistakes. In this paper we improve this ratio to $\frac{3}{2}|F|$ mistakes using a pivot-based method. We also propose novel 2-approximations for BTT. Using a recent result on approximating the bad triangle cover LP, we obtain an $(2+\epsilon)$ approximation in time almost equal to the time needed to find a maximal set of edge-disjoint bad triangles (which would give a standard 3-approximation). Additionally, several inapproximability results are provided. For general signed graphs, a better than 2-approximation is unlikely as our problem can be used to approximate vertex cover. For complete signed graphs, it is NP-hard to approximate with factor better than $\frac{2137}{2136}$. This result also holds for several other related problems.