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Lower Bounds for Symmetric Circuits for the Determinant

Anuj Dawar, Gregory Wilsenach

TL;DR

The paper proves an exponential lower bound for Alt_n × Alt_n-symmetric arithmetic circuits computing the determinant over fields of characteristic zero, advancing prior results that used weaker symmetry. It develops a general framework combining a support theorem and restricted bijection games, together with a CFI-based biadjacency construction to realize the bound and analyze symmetry as a computational resource. The approach yields a principled method to study trade-offs between symmetry and circuit size and provides new tools that apply to a broader class of symmetric computations beyond the determinant. These contributions deepen our understanding of how invariance constraints shape the computability of algebraic objects and open avenues for exploring symmetry as a resource in arithmetic circuit complexity.

Abstract

Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is that the circuits which take a matrix input are unchanged by a permutation applied simultaneously to the rows and columns of the matrix. Under such restrictions we have polynomial-size circuits for computing the determinant but no subexponential size circuits for the permanent. Here, we consider a more stringent symmetry requirement, namely that the circuits are unchanged by arbitrary even permutations applied separately to rows and columns, and prove an exponential lower bound even for circuits computing the determinant. The result requires substantial new machinery. We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games. These are applied to the determinant problem with a novel construction of matrices that are bi-adjacency matrices of graphs based on the CFI construction. Our general framework opens the way to exploring a variety of symmetry restrictions and studying trade-offs between symmetry and other resources used by arithmetic circuits.

Lower Bounds for Symmetric Circuits for the Determinant

TL;DR

The paper proves an exponential lower bound for Alt_n × Alt_n-symmetric arithmetic circuits computing the determinant over fields of characteristic zero, advancing prior results that used weaker symmetry. It develops a general framework combining a support theorem and restricted bijection games, together with a CFI-based biadjacency construction to realize the bound and analyze symmetry as a computational resource. The approach yields a principled method to study trade-offs between symmetry and circuit size and provides new tools that apply to a broader class of symmetric computations beyond the determinant. These contributions deepen our understanding of how invariance constraints shape the computability of algebraic objects and open avenues for exploring symmetry as a resource in arithmetic circuit complexity.

Abstract

Dawar and Wilsenach (ICALP 2020) introduce the model of symmetric arithmetic circuits and show an exponential separation between the sizes of symmetric circuits for computing the determinant and the permanent. The symmetry restriction is that the circuits which take a matrix input are unchanged by a permutation applied simultaneously to the rows and columns of the matrix. Under such restrictions we have polynomial-size circuits for computing the determinant but no subexponential size circuits for the permanent. Here, we consider a more stringent symmetry requirement, namely that the circuits are unchanged by arbitrary even permutations applied separately to rows and columns, and prove an exponential lower bound even for circuits computing the determinant. The result requires substantial new machinery. We develop a general framework for proving lower bounds for symmetric circuits with restricted symmetries, based on a new support theorem and new two-player restricted bijection games. These are applied to the determinant problem with a novel construction of matrices that are bi-adjacency matrices of graphs based on the CFI construction. Our general framework opens the way to exploring a variety of symmetry restrictions and studying trade-offs between symmetry and other resources used by arithmetic circuits.

Paper Structure

This paper contains 21 sections, 20 theorems, 14 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Groups
  4. Matrices
  5. Graphs
  6. Circuits
  7. Symmetric Circuits
  8. Group Actions and Symmetric Functions
  9. Symmetric Circuits
  10. Games and Supports
  11. Bijection Games on Indexed function
  12. Supports
  13. Playing Games on Circuits
  14. Bounds on Supports
  15. Lower Bound for the Determinant
  16. ...and 6 more sections

Key Result

Theorem 2.1

There is a winning strategy for the cop player with $k$ cops if, and only if, the tree-width of $\Gamma$ is at most $k-1$.

Figures (4)

  • Figure 1: A gadget in $\hat{\Gamma}$ corresponding to vertex $v$ with neighbours $x,y,z$
  • Figure 2: The gadget from Figure \ref{['fig:gadget']} with $x_0$ and $y_0$ removed
  • Figure 3: An alternating cycle between two matchings in $\mu(F,f)$ and $\mu(F,g)$ where $f$ and $g$ differ on the edge $\{v,x\}$.
  • Figure 4: A gadget in $X(\Gamma)$ corresponding to vertex $v$ with incident edges $f,g,h$

Theorems & Definitions (47)

  • Theorem 2.1: SEYMOUR199322
  • Definition 2.2: Circuit
  • Definition 3.1
  • Example 3.2
  • Definition 3.3: Circuit Automorphism
  • Definition 3.4: Symmetric Circuit
  • Proposition 3.5
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • ...and 37 more