Beyond Worst-Case Analysis for Symbolic Computation: Root Isolation Algorithms

2025-06-04

AI Summary

This paper addresses the fundamental symbolic computation problem of real root isolation for univariate integer polynomials. It introduces smoothed analysis—previously absent in symbolic computation—to bridge the gap between practical efficiency and worst-case complexity bounds. Methodologically, it integrates Descartes’ rule of signs, Sturm sequences, the hybrid ANewDsc algorithm, and sparse polynomial techniques to establish a smoothed complexity framework. Key contributions include: (i) the first proof that Descartes’ method achieves quasilinear bit complexity in both expectation and under smoothing; (ii) the first non-worst-case theoretical guarantees for Sturm’s method, ANewDsc, and sparse symbolic solvers; and (iii) a systematic explanation of why classical algorithms consistently outperform their worst-case bounds in practice. This work fills a long-standing gap in the average-case and smoothed complexity theory of symbolic computation.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We introduce beyond-worst-case analysis into symbolic computation. This is an extensive field which almost entirely relies on worst-case bit complexity, and we start from a basic problem in the field: isolating the real roots of univariate polynomials. This is a fundamental problem in symbolic computation and it is arguably one of the most basic problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, most available results in literature either focus on worst-case analysis in the bit complexity model or simply provide experimental benchmarking without any theoretical justifications of the observed results. We aim to address the discrepancy between practical performance of root isolation algorithms and prescriptions of worst-case complexity theory: We develop a smoothed analysis framework for polynomials with integer coefficients to bridge this gap. We demonstrate (quasi-)linear (expected and smoothed) complexity bounds for Descartes algorithm, that is one most well know symbolic algorithms for isolating the real roots of univariate polynomials with integer coefficients. Our results explain the surprising efficiency of Descartes solver in comparison to sophisticated algorithms that have superior worst-case complexity. We also analyse the Sturm solver, ANewDsc a symbolic-numeric algorithm that combines Descartes with Newton operator, and a symbolic algorithm for sparse polynomials.

Problem

• Analyzing real root isolation beyond worst-case complexity
• Bridging gap between practical performance and theoretical complexity
• Evaluating efficiency of Descartes algorithm versus sophisticated methods

Innovation

• Smoothed analysis framework for polynomials
• Quasi-linear complexity for Descartes algorithm
• Comparison of Descartes with sophisticated algorithms