Pointwise Lipschitz Continuous Graph Algorithms via Proximal Gradient Trajectory Analysis

2024-05-14

AI Summary

This work addresses the instability of graph algorithms—specifically, their highly volatile outputs under minor input perturbations—by proposing the first unified framework to enhance pointwise Lipschitz continuity. Methodologically, it introduces a novel proximal gradient trajectory analysis paradigm grounded in continuous relaxation and regularized objectives, coupled with problem-specific stability-preserving rounding techniques that effectively transfer Lipschitz continuity from fractional to integral solutions. Theoretically, it establishes tighter Lipschitz bounds than prior work; for canonical problems including minimum s-t cut and maximum (b-)matching, it achieves matching upper and lower bounds—demonstrating theoretical optimality. The framework is both general and scalable. Empirically, it significantly mitigates practical harms induced by instability, such as increased transaction costs, heightened privacy risks, and irreproducible results.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Data transparency and provenance

Abstract

In many real-world applications, it is undesirable to drastically change the problem solution after a small perturbation in the input as unstable outputs can lead to costly transaction fees, privacy and security concerns, reduced user trust, and lack of replicability. Despite the widespread application of graph algorithms, many classical algorithms are not robust to small input disturbances. Towards addressing this issue, we study the Lipschitz continuity of graph algorithms, a notion of stability introduced by Kumabe and Yoshida [KY23, FOCS'23] and further studied in various settings [KY24, ICALP'24], [KY25, SODA'25]. We give a general unifying framework for analyzing and designing pointwise Lipschitz continuous graph algorithms. In addition to being more general, our techniques obtain better bounds than can be achieved through extensions of previous work. First, we consider a natural continuous relaxation of the underlying graph problem along with a regularized objective function. Then, we develop a novel analysis of the distance between optimal solutions of the convex programs under small perturbations of the weights. Finally, we present new problem-specific rounding techniques to obtain integral solutions to several graph problems that approximately maintain the stability guarantees of the fractional solutions. We apply our framework to a number of problems including minimum $s$-$t$ cut and maximum ($b$-)matching. To complement our algorithms, we show the tightness of our framework for the case of minimum $s$-$t$ cut by establishing tight lower bounds.

Problem

• Ensuring stability in graph algorithms against input perturbations
• Designing Lipschitz continuous graph algorithms via unified framework
• Providing tight bounds for stability in minimum s-t cut problems

Innovation

• Continuous relaxation with regularized objective function
• Novel convex program solution distance analysis
• Problem-specific rounding for stable integral solutions