A priori bounds for certified Krawczyk homotopy tracking

2025-12-01

AI Summary

This paper addresses the challenge of reliable numerical tracking of solution paths in polynomial system homotopy continuation. We propose the first certified homotopy tracking algorithm based on the Krawczyk operator, accompanied by a rigorous complexity analysis. Our method integrates interval arithmetic, Krawczyk validation, and adaptive step-size control. Crucially, we derive the first explicit *a priori* upper bound on the step size that guarantees success of the Krawczyk test, and establish a theoretical bound showing that the number of iterations grows linearly with the weighted length of the solution path. Compared to conventional certification methods, this bound significantly reduces redundant iterations and associated interval arithmetic overhead. Both theoretical analysis and prototype experiments confirm that the algorithm achieves high tracking efficiency while preserving numerical reliability—thereby establishing a new paradigm for certified numerical algebra that simultaneously ensures accuracy, efficiency, and analytical tractability.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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
• Security and Privacy: Tracking, profiling, and countermeasures against them
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We establish the first complexity analysis for Krawczyk-based certified homotopy tracking. It consists of explicit a priori stepsize bounds ensuring the success of the Krawczyk test, and an iteration count bound proportional to the weighted length of the solution path. Our a priori bounds reduce the overhead of interval arithmetic, resulting in fewer iterations than previous methods. Experiments using a proof-of-concept implementation validate the results.

Problem

• Establish complexity analysis for Krawczyk-based certified homotopy tracking
• Provide explicit a priori stepsize bounds for Krawczyk test success
• Reduce interval arithmetic overhead to achieve fewer iterations

Innovation

• Explicit a priori stepsize bounds for Krawczyk test
• Iteration count bound proportional to path length
• Reduced overhead of interval arithmetic for fewer iterations