[Paper Review] Search problems in algebraic complexity, GCT, and hardness of generator for invariant rings

Ankit Garg, Christian Ikenmeyer+4 more|arXiv (Cornell University)|Oct 2, 2019

Commutative Algebra and Its Applications22 references4 citations

TL;DR

This paper disproves Mulmuley's conjecture that invariant rings of SLₙ(ℂ)-representations admit polynomial-sized succinct encodings by constructing explicit counterexamples using hyperpfaffians. It establishes that computing such generators is VNP-complete, implying that a polynomial-sized encoding would collapse VP to VNP under standard complexity assumptions.

ABSTRACT

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of $\SL_n(\C)$-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions). We develop a general framework, denoted \emph{algebraic circuit search problems}, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings.

Motivation & Objective

• To investigate the existence of polynomial-sized succinct encodings for generators of invariant rings under SLₙ(ℂ)-representations.
• To challenge Mulmuley's conjecture in Geometric Complexity Theory (GCT) that such encodings always exist.
• To establish connections between algebraic circuit search problems, invariant theory, and fundamental complexity classes like VP and VNP.
• To demonstrate that computing generators for certain invariant rings is as hard as solving VNP-complete problems.
• To provide a general framework—algebraic circuit search problems—for unifying problems in algebraic complexity and computational invariant theory.

Proposed method

• Introduces the formal framework of algebraic circuit search problems to unify problems in algebraic complexity and invariant theory.
• Uses the hyperpfaffian polynomial Pfₖ,ₙ as a key invariant under SL₂ₖₙ(ℂ) action on ⊗²ᵏℂ²ᵏⁿ.
• Shows that Pfₖ,ₙ is VNP-complete for even k ≥ 2, using a projection from a tensor product to the permanent.
• Constructs a linear parametrization of a tensor p such that ⟨v, p⊗ⁿ⟩ equals the permanent up to a constant.
• Proves that any polynomial-sized succinct encoding of the invariant ring generators would yield a polynomial-size circuit for Pfₖ,ₙ.
• Applies the VNP-completeness of Pfₖ,ₙ to show that such an encoding implies VP = VNP, contradicting standard complexity assumptions.

Experimental results

Research questions

• RQ1 Does every invariant ring of an SLₙ(ℂ)-representation admit a polynomial-sized succinct encoding, as conjectured by Mulmuley?
• RQ2 Can the hyperpfaffian polynomial Pfₖ,ₙ be computed by a polynomial-size algebraic circuit, given its role in generating invariants?
• RQ3 Is the problem of computing generators for invariant rings equivalent in complexity to VNP-complete problems?
• RQ4 Can the permanent be projected from a tensor invariant under SL₂ₖₙ(ℂ), and what does this imply for circuit complexity?
• RQ5 What are the implications of a polynomial-sized encoding of invariants for the VP vs. VNP problem?

Key findings

• The paper constructs explicit counterexamples to Mulmuley's conjecture, showing that not all SLₙ(ℂ)-invariant rings admit polynomial-sized succinct encodings.
• For even k ≥ 2, the hyperpfaffian Pfₖ,ₙ is VNP-complete, establishing its high computational complexity.
• The evaluation of Pfₖ,ₙ at a linearly parametrized tensor p yields the d×d permanent up to a nonzero scalar, proving a projection from Pfₖ,ₙ to Perₙ.
• The space of homogeneous invariants of degree n under the SL₂ₖₙ(ℂ) action on ⊗²ᵏℂ²ᵏⁿ is 1-dimensional and spanned by Pfₖ,ₙ.
• Any polynomial-sized succinct encoding of the invariant ring generators would imply a polynomial-size circuit for Pfₖ,ₙ, which would collapse VP to VNP.
• Thus, under standard complexity assumptions (VP ≠ VNP), no such polynomial-sized encoding exists for these invariant rings.