The Bad Triangle Transversal (BTT) problem asks for the smallest set of edges that need to be removed from a given signed graph, so that the resulting graph does not have a bad triangle. Here, a bad triangle is a triangle with exactly one negative edge. Several 2-approximations for BTT are proposed in this paper. On the hardness side, we show that BTT is NP-hard to approximate with factor better than $\frac{2137}{2136}$ on complete graphs. Our reduction also works for Correlation Clustering (CC), the Cluster Deletion problem (CD) and the Minimum Strong Triadic Closure problem (MinSTC). Lastly, we show that the BTT and CC optima are within a factor of 3/2 in complete graphs, by describing a pivot procedure that transforms transversals into clusters.

翻译：坏三角形横贯（BTT）问题要求从给定的符号图中移除最少的边，使得结果图中不存在坏三角形。此处，坏三角形指恰好包含一条负边的三角形。本文提出了BTT问题的几个2-近似算法。在难解性方面，我们证明在完全图上，BTT问题是NP-难问题，且近似因子难以优于$\frac{2137}{2136}$。我们的归约同样适用于相关聚类（CC）、簇删除问题（CD）以及最小强三元闭包问题（MinSTC）。最后，通过描述一种将横贯转化为簇的枢轴过程，我们证明在完全图上，BTT问题与CC问题的最优解相差不超过因子3/2。