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Completeness classes in algebraic complexity theory

Peter Bürgisser

TL;DR

Valiant's framework recasts a central computer science question into algebraic geometry and invariant theory, introducing the complexity classes $VP$ and $VNP$ and establishing determinant and permanent as canonical complete problems. The paper situates arithmetic circuits as the computation model and develops a robust reduction system via substitutions (projections), explores completeness landscapes (determinant in $VBP$, permanent in $VNP$), and connects to generating functions of graph properties and determinantal complexity. It then surveys robustness, factorization, and intermediate complexity, introduces second-generation classes (constant-free, non-bounded degree, VPSPACE), and discusses derandomization of PIT and restricted models. The Geometric Complexity Theory program is presented as a blueprint to attack VP vs. VNP via orbit closures, representation-theoretic obstructions, and symmetry considerations, outlining both the promises and the obstacles ahead. Together, these perspectives form a comprehensive foundation for the algebraic approach to fundamental complexity separations and their geometric incarnations.

Abstract

The purpose of this overview is to explain the enormous impact of Les Valiant's eponymous short conference contribution from 1979 on the development of algebraic complexity.

Completeness classes in algebraic complexity theory

TL;DR

Valiant's framework recasts a central computer science question into algebraic geometry and invariant theory, introducing the complexity classes and and establishing determinant and permanent as canonical complete problems. The paper situates arithmetic circuits as the computation model and develops a robust reduction system via substitutions (projections), explores completeness landscapes (determinant in , permanent in ), and connects to generating functions of graph properties and determinantal complexity. It then surveys robustness, factorization, and intermediate complexity, introduces second-generation classes (constant-free, non-bounded degree, VPSPACE), and discusses derandomization of PIT and restricted models. The Geometric Complexity Theory program is presented as a blueprint to attack VP vs. VNP via orbit closures, representation-theoretic obstructions, and symmetry considerations, outlining both the promises and the obstacles ahead. Together, these perspectives form a comprehensive foundation for the algebraic approach to fundamental complexity separations and their geometric incarnations.

Abstract

The purpose of this overview is to explain the enormous impact of Les Valiant's eponymous short conference contribution from 1979 on the development of algebraic complexity.
Paper Structure (37 sections, 52 theorems, 65 equations, 4 figures)

This paper contains 37 sections, 52 theorems, 65 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Foundations
  3. Arithmetic circuits
  4. Valiant's complexity classes
  5. Reduction via substitution
  6. Circuits for the determinant
  7. Completeness of the determinant
  8. The class $\mathrm{VNP}$
  9. Completeness of the permanent
  10. Generating functions of graph properties
  11. Determinantal complexity
  12. Properties of complexity classes
  13. Robustness
  14. Complexity of factors
  15. Families of intermediate complexity
  16. ...and 22 more sections

Key Result

Proposition 2.2

Let $\Phi$ be an arithmetic circuit $\Phi$ of size $s$ and let $d\ge 1$. Then there is a homogenous arithmetic circuit of size $O(sd^2)$, which computes all the homogeneous components of all the polynomials $\hat{\Phi}_{v}$ computed by $\Phi$.

Figures (4)

  • Figure 1: An arithmetic circuit of size 7, depth 6 and degree 3 computing $-x^3+xy+y^2-1$.
  • Figure 2: Iff-coupling of the edges $c=(u,v)$ and $c'=(u',v')$.
  • Figure 3: The rosette $R(4)$.
  • Figure 4: The digraph encoding the monomial $x_1 x_2 x_3^2 x_4^3$.

Theorems & Definitions (99)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Example 2.1
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 89 more