Dual-Bounded Nonlinear Optimal Transport for Size Constrained Min Cut Clustering

2025-01-30

AI Summary

For the size-constrained minimum cut (SCMC) problem, existing methods suffer from high computational cost, complex modeling, and coarse solutions. This paper introduces the first formulation of SCMC as a dual-bounded nonlinear optimal transport problem and proposes DNF—a parameter-free, highly stable solving framework. DNF integrates Frank–Wolfe iterative optimization, dual-constraint modeling, and nonlinear optimal transport theory. Theoretically, it achieves an $O(1/t)$ convergence rate under convex smooth settings; empirically, it attains $O(1/sqrt{t})$ acceleration. On SCMC benchmarks, DNF substantially outperforms state-of-the-art methods: it yields lower objective loss, higher clustering accuracy, and fastest convergence—while requiring no hyperparameter tuning and exhibiting strong robustness.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Min cut is an important graph partitioning method. However, current solutions to the min cut problem suffer from slow speeds, difficulty in solving, and often converge to simple solutions. To address these issues, we relax the min cut problem into a dual-bounded constraint and, for the first time, treat the min cut problem as a dual-bounded nonlinear optimal transport problem. Additionally, we develop a method for solving dual-bounded nonlinear optimal transport based on the Frank-Wolfe method (abbreviated as DNF). Notably, DNF not only solves the size constrained min cut problem but is also applicable to all dual-bounded nonlinear optimal transport problems. We prove that for convex problems satisfying Lipschitz smoothness, the DNF method can achieve a convergence rate of (mathcal{O}(frac{1}{t})). We apply the DNF method to the min cut problem and find that it achieves state-of-the-art performance in terms of both the loss function and clustering accuracy at the fastest speed, with a convergence rate of (mathcal{O}(frac{1}{sqrt{t}})). Moreover, the DNF method for the size constrained min cut problem requires no parameters and exhibits better stability.

Problem

• Minimum Cut
• Algorithm Optimization
• Graph Partitioning

Innovation

• Minimum Cut
• DNF Method
• Transmission Problem Transformation