  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.
