A simple inverse power method for balanced graph cut
Sihong Shao, Chuan Yang
TL;DR
This work tackles the NP-hard balanced graph cut by replacing the challenging inner subproblem of the inverse power method with a closed-form, analytically solvable subproblem, and coupling it with a boundary-detected subgradient to guarantee local convergence. The authors introduce the simple inverse power (SIP) method and its θ-variant SIP_θ, plus SIP-perturb as a local breakout strategy, yielding strong computational efficiency and competitive solution quality on Cheeger and Sparsest cuts. They establish rigorous convergence properties and demonstrate substantial speedups over IP and improved performance versus Gurobi in extensive G-set experiments. The approach leverages a novel continuous formulation via Lovász extensions, enabling efficient iteration and potential extensions to broader combinatorial–continuous mappings with practical impact in graph clustering and partitioning tasks.
Abstract
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 $θ$-balanced cut which reduces to the balanced cut when $θ=1$. When $\mathbf{SIP}$ reaches its local optimum, we seamlessly transfer to solve the $θ$-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 $θ$. 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.