Given a signed graph, the bad triangle transversal (BTT) problem asks to find the smallest number of edges that need to be removed such that the remaining graph does not have a triangle with exactly one negative edge (a bad triangle). We propose novel 2-approximations for this problem, which are much simpler and faster than a folklore adaptation of the 2-approximation by Krivelevich for finding a minimum triangle transversal in unsigned graphs. One of our algorithms also works for weighted BTT and for approximately optimal feasible solutions to the bad triangle cover LP. Using a recent result on approximating the bad triangle cover LP, we obtain a $(2+ε)$ 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 complete signed graphs, we show that BTT is NP-hard to approximate with factor better than $\frac{2137}{2136}$. Our reduction also implies the same hardness result for related problems such as correlation clustering (cluster editing), cluster deletion and the min. strong triadic closure problem. On complete signed graphs, BTT is closely related to correlation clustering. We show that the correlation clustering optimum is at most $3/2$ times the BTT optimum, by describing a pivot procedure that transforms BTT solutions into clusters. This improves a result by Veldt, which states that their ratio is at most two.

翻译：给定一个带符号图，坏三角形横贯集（BTT）问题要求找出需要移除的最小边数，使得剩余图中不存在恰好包含一条负边的三角形（即坏三角形）。我们针对该问题提出了新颖的2-近似算法，相较于Krivelevich针对无符号图中最小三角形横贯集问题的2-近似算法的经典改编版本，本文算法更为简洁高效。其中一种算法还可推广至加权BTT问题及坏三角形覆盖线性规划的近似最优可行解。借助近期关于坏三角形覆盖线性规划近似求解的研究成果，我们获得了时间复杂度几乎等同于寻找极大边不相交坏三角形集合（该方法可产生标准3-近似解）的$(2+ε)$近似算法。此外，本文还提供了若干不可近似性结果。对于完全带符号图，我们证明BTT问题在近似比优于$\frac{2137}{2136}$时具有NP困难性。该归约过程同时表明相关问题（如相关聚类、聚类编辑、聚类删除及最小强三元闭包问题）具有相同的困难性下界。在完全带符号图中，BTT问题与相关聚类密切相关。我们通过描述将BTT解转化为聚类结果的枢轴操作过程，证明相关聚类最优值至多为BTT最优值的$3/2$倍，该结果改进了Veldt提出的二者比值至多为2的结论。