An instance of the Connected Maximum Cut problem consists of an undirected graph $G=(V,E)$ and the goal is to find a subset of vertices $S\subseteq V$ that maximizes the number of edges in the cut $\delta (S)$ such that the induced graph $G[S]$ is connected. We present the first non-trivial $\mathrm{\Omega }(\frac{1}{\log n})$ approximation algorithm for the Connected Maximum Cut problem in general graphs using novel techniques. We then extend our algorithm to edge weighted case and obtain a poly-logarithmic approximation algorithm. Interestingly, in contrast to the classical Max-Cut problem that can be solved in polynomial time on planar graphs, we show that the Connected Maximum Cut problem remains NP-hard on unweighted, planar graphs. On the positive side, we obtain a polynomial time approximation scheme for the Connected Maximum Cut problem on planar graphs and more generally on bounded genus graphs.

Connected Maximum Cut问题的一个实例由一个无向图组成$G=（V，Ë）$ 目标是找到顶点的子集 $\mathrm{小号}\subseteq V$ 最大化切割中的边数 $\delta （\mathrm{小号}）$ 这样诱导图 $G[\mathrm{小号}]$已连接。我们提出第一个不平凡的$\mathrm{\Omega }（\frac{\mathrm{1个}}{\mathrm{日志}ñ}）$通用图的连通最大割问题的近似算法，采用了新颖的技术。然后，我们将算法扩展到边缘加权情况，并获得一个多对数近似算法。有趣的是，与可以在多项式时间内在平面图上求解的经典Max-Cut问题相反，我们证明了Connected Maximum Cut问题在未加权的平面图上仍然是NP-hard。从积极的方面来说，我们为平面图上以及更普遍的有界属图上的连通最大割问题获得了多项式时间逼近方案。