The existing inverse power ($\mathbf{IP}$) method for solving the balanced graph cut lacks local convergence and its inner subproblem requires a nonsmooth convex solver. To address these issues, we develop a simple inverse power ($\mathbf{SIP}$) method using a novel equivalent continuous formulation of the balanced graph cut, and its inner subproblem allows an explicit analytic solution, which is the biggest advantage over $\mathbf{IP}$ and constitutes the main reason why we call it $\mathit{simple}$. By fully exploiting the closed-form of the inner subproblem solution, we design a boundary-detected subgradient selection with which $\mathbf{SIP}$ is proved to be locally converged. We show that $\mathbf{SIP}$ is also applicable to a new ternary valued $\theta$-balanced cut which reduces to the balanced cut when $\theta=1$. When $\mathbf{SIP}$ reaches its local optimum, we seamlessly transfer to solve the $\theta$-balanced cut within exactly the same iteration algorithm framework and thus obtain $\mathbf{SIP}$-$\mathbf{perturb}$ -- an efficient local breakout improvement of $\mathbf{SIP}$, which transforms some ``partitioned" vertices back to the ``un-partitioned" ones through the adjustable $\theta$. Numerical experiments on G-set for Cheeger cut and Sparsest cut demonstrate that $\mathbf{SIP}$ is significantly faster than $\mathbf{IP}$ while maintaining approximate solutions of comparable quality, and $\mathbf{SIP}$-$\mathbf{perturb}$ outperforms $\mathtt{Gurobi}$ in terms of both computational cost and solution quality.

翻译：现有的求解平衡图割的逆幂（$\mathbf{IP}$）方法缺乏局部收敛性，且其内部子问题需要一个非光滑凸求解器。为解决这些问题，我们利用平衡图割的一种新颖等价连续公式，发展了一种简单逆幂（$\mathbf{SIP}$）方法，其内部子问题允许一个显式解析解，这是相对于$\mathbf{IP}$的最大优势，也是我们称其为$\mathit{简单}$的主要原因。通过充分利用内部子问题解的闭式形式，我们设计了一种边界检测次梯度选择方法，并证明了$\mathbf{SIP}$在该方法下具有局部收敛性。我们证明了$\mathbf{SIP}$也适用于一种新的三元值$\theta$-平衡割，当$\theta=1$时该割即退化为平衡割。当$\mathbf{SIP}$达到其局部最优解时，我们无缝地转换到在同一迭代算法框架内求解$\theta$-平衡割，从而得到$\mathbf{SIP}$-$\mathbf{perturb}$——这是$\mathbf{SIP}$的一种高效局部突破改进，它通过可调节的$\theta$将一些“已划分”的顶点转换回“未划分”状态。在G-set数据集上针对Cheeger割和Sparsest割进行的数值实验表明，$\mathbf{SIP}$在保持近似解质量相当的同时，速度显著快于$\mathbf{IP}$；并且$\mathbf{SIP}$-$\mathbf{perturb}$在计算成本和求解质量两方面均优于$\mathtt{Gurobi}$。