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If VNP is hard, then so are equations for it

Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, Anamay Tengse

TL;DR

The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

Abstract

Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

If VNP is hard, then so are equations for it

TL;DR

The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

Abstract

Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

Paper Structure

This paper contains 15 sections, 4 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Complexity of Equations for classes of polynomials
  3. Our results
  4. An overview of the proof
  5. Details of the proof.
  6. Preliminaries
  7. Notation
  8. Some basic definitions
  9. Proof of the main theorem
  10. Building monomials from exponent vectors
  11. Indexing Combinatorial Designs Algebraically
  12. Reed-Solomon based combinatorial designs
  13. The VNP-Succinct-KI generator
  14. Putting it all together
  15. Discussion and Open Problems

Key Result

theorem 1.1

For every constant $c > 0$, there is a polynomial family $\{P_{N, c}\} \in \mathsf{VP}_\mathbb{Q}$For a field $\mathbb{F}$, $\mathsf{VP}_{\mathbb{F}}$ denotes the class $\mathsf{VP}$ where the coefficients of the polynomials are from the field $\mathbb{F}$. Similarly, $\mathsf{VNP}_{\mathbb{F}}$ den Here, $\overline{\operatorname{coeff}}(f)$ denotes the coefficient vector of a polynomial $f$.

Theorems & Definitions (16)

  • theorem 1.1: CKRST20
  • theorem 1.1: Conditional Hardness of Equations for VNP
  • Remark 1
  • definition 2.1: Algebraic circuits
  • definition 2.2: $\mathsf{VP}$ and $\mathsf{VNP}$
  • definition 2.3: Equations for a class
  • definition 2.4: Hitting Set Generator (HSG)
  • definition 2.5: Succinct Hitting Sets for a class of polynomials GKSS17FSV18
  • definition 2.6: Combinatorial designs
  • Lemma 2.7: HSG from Hardness KI04
  • ...and 6 more