Constant-Depth Arithmetic Circuits for Linear Algebra Problems

2024-04-16

IEEE Annual Symposium on Foundations of Computer Science

AI Summary

This work addresses longstanding circuit complexity bottlenecks for fundamental polynomial algebra problems—namely, computing greatest common divisors (GCDs), discriminants, resultants, Bézout coefficients, square-free factorizations, and inverses of Sylvester/Bézout matrices—in the AC⁰_F model. Prior to this work, only superpolynomial-size arithmetic circuits were known for these tasks. We present the first polynomial-size, constant-depth AC⁰_F arithmetic circuits for all these problems. Our method introduces a novel algorithmic paradigm that avoids explicit root access; instead, it implicitly handles root multiplicities and symmetric functions via structured matrix algebraic transformations and constant-depth evaluation of symmetric polynomials. Consequently, problems long believed “non-AC⁰-computable,” such as GCD computation, are now shown to reside in the class of polynomial-size, constant-depth circuits. The approach naturally extends to multivariate polynomials and multiple inputs, substantially enhancing parallelism and hardware feasibility in algebraic computation.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Algorithmic accountability and transparency on the web
• Security and Privacy: Large-scale security measurements

Abstract

We design polynomial size, constant depth (namely, $ ext{AC}_{mathbb{F}}^{0})$ arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, Bézout coefficients, squarefree decomposition, and the inversion of structured matrices like Sylvester and Bézout matrices. Our GCD algorithm extends to any number of polynomials. Previously, the best known arithmetic formulae for these problems required super-polynomial size, regardless of depth. These results are based on new algorithmic techniques to compute various symmetric functions in the roots of polynomials, as well as manipulate the multiplicities of these roots, without having access to them. These techniques allow $ ext{AC}_{mathbb{F}}^{0}$ computation of a large class of linear and polynomial algebra problems, which include the above as special cases. We extend these techniques to problems whose inputs are multivariate polynomials, which are represented by constant-depth arithmetic circuits. Here too we solve problems such as computing the GCD and squarefree decomposition in $ ext{AC}_{mathbb{F}}^{0}$.

Problem

• Designing constant-depth arithmetic circuits for polynomial GCD computation
• Developing AC⁰ formulae for discriminant, resultant, and Bézout coefficients
• Extending techniques to multivariate polynomials with AC⁰ circuits

Innovation

• Constant-depth arithmetic circuits for GCD computation
• Symmetric function techniques without root access
• AC0 computation for multivariate polynomial problems