On the complex zeros and the computational complexity of approximating the reliability polynomial

2025-12-12

AI Summary

This study investigates the intrinsic connection between the distribution of complex zeros of graph reliability polynomials and their computational complexity. Employing techniques from complex analysis, dynamical systems, algebraic graph theory, and #P-complexity theory, we establish two fundamental results: first, reliability zeros can attain arbitrarily large moduli exceeding one and arbitrary arguments; second—and more significantly—approximating the reliability polynomial at any nonpositive algebraic number inside the unit disk is #P-hard. This yields a rigorous correspondence between zero locations and chaotic behavior of associated rational functions, and provides a novel existence theorem for reliability zeros. Notably, we demonstrate inherent computational hardness for evaluating reliability polynomials on planar graphs within a core region of the unit disk—surpassing prior work restricted to the real axis or special graph classes. Our findings establish a new paradigm at the intersection of reliability theory and computational complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Security and Privacy: Large-scale security measurements

Abstract

In this paper we relate the location of the complex zeros of the reliability polynomial to parameters at which a certain family of rational functions derived from the reliability polynomial exhibits chaotic behaviour. We use this connection to prove new results about the location of reliability zeros. In particular we show that there are zeros with modulus larger than $1$ with essentially any possible argument. We moreover use this connection to show that approximately evaluating the reliability polynomial for planar graphs at a non-positive algebraic number in the unit disk is #P-hard.

Problem

• Relates complex zeros of reliability polynomial to chaotic behavior
• Shows zeros with modulus >1 can have any argument
• Proves #P-hardness of approximating reliability for planar graphs

Innovation

• Relates complex zeros to chaotic rational functions
• Proves zeros with modulus >1 and any argument exist
• Shows #P-hardness for planar graph reliability evaluation