Determinant versus permanent

  title={Determinant versus permanent},
  author={Manindra Agrawal},
We study the problem of expressing permanents of matrices as determinants of (possibly larger) matrices. This problem has close connections to the complexity of arithmetic computations: the complexities of computing permanent and determinant roughly correspond to arithmetic versions of the classes NP and P respectively. We survey known results about their relative complexity and describe two recently developed approaches that might lead to a proof of the conjecture that the permanent can only… 

Permanent vs. Determinant

A major problem in theoretical computer science is the Permanent vs. Determinant problem. It asks: given an n by n matrix of indeterminates A = (ai,j) and an m by m matrix B with entries that are


The Permanent versus Determinant problem is the following: Given an n × n matrix X of indeterminates over a field of characteristic different from two, find the smallest matrix M whose coefficients

A quadratic lower bound for the permanent and determinant problem over any characteristic ≠ 2

The Mignon-Ressayre quadratic lower bound m=Ω(n2) for fields of characteristic 0 is extended to all Fields of characteristic ≠2.

A method for deriving lower bounds for the complexity of monotone arithmetic circuits computing real polynomials

This work suggests a method for deriving lower bounds for the complexity of polynomials with positive real coefficients implemented by circuits of functional elements over the monotone arithmetic

Progress on Polynomial Identity Testing - II

  • Nitin Saxena
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 2013
The area of algebraic complexity theory is surveyed, with the focus being on the problem of polynomial identity testing (PIT), and the key ideas are discussed.

An identity between the determinant and the permanent of Hessenberg-type matrices

In this short note we provide an extension of the notion of Hessenberg matrix and observe an identity between the determinant and the permanent of such matrices. The celebrated identity due to Gibson

Determinantal Complexities and Field Extensions

This paper states that the determinantal complexity of the permanent polynomial is the smallest size of a matrix M whose entries are linear polynomials of x i ’s over \(\mathbb{F}\), such that P=det (M) as polynmials in \(\mathBB{F}[x_1, \dots, x_n]\).

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

This work studies the complexity of two special but natural cases of identity testing—first is a case of depth-3 PIT, the other ofdepth-4 PIT; generalize this technique by combining it with a generalized Chinese remaindering to solve these two problems (over any field).

Sparse instances of hard problems

Under the hypothesis that coNP is not in NP/poly, the results imply surprisingly tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs.

Affine projections of polynomials

  • N. Kayal
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 2011
The intention of this paper is to understand the complexity of the corresponding computational problem: given polynomials f and g find A and b such that f = g(Ax + b), if such an (A,b) exists.



A quadratic bound for the determinant and permanent problem

The determinantal complexity of a polynomial f is defined here as the minimal size of a matrix M with affine entries such that f = det M. This function gives a minoration of the more traditional size

Determinant: Combinatorics, Algorithms, and Complexity

It is proved that the complexity class GapL characterizes the complexity of computing the determinant of matrices over the integers, and a direct proof of this characterization is presented.

Multi-linear formulas for permanent and determinant are of super-polynomial size

It is proved that any multi-linear arithmetic formula for the permanent or the determinant of an n x n matrix is of size super-polynomial in n.

Lower bounds on arithmetic circuits via partial derivatives

  • N. NisanA. Wigderson
  • Mathematics, Computer Science
    Proceedings of IEEE 36th Annual Foundations of Computer Science
  • 1995
A new technique is described for obtaining lower bounds on restricted classes of non-monotone arithmetic circuits, based on the linear span of their partial derivatives, for multivariate polynomials.

Depth-3 arithmetic formulae over fields of characteristic zero

  • Amir ShpilkaA. Wigderson
  • Mathematics, Computer Science
    Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317)
  • 1999
This paper proves near quadratic lower bounds for depth-3 arithmetic formulae over fields of characteristic zero for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant, and gets the first non-trivial lower bound for computing polynomials of constant degree.

Complete Problems for Valiant's Class of qp-Computable Families of Polynomials

We prove that the families matrix powering, iterated matrix product, and adjoint matrix are VQP-complete, where VQP denotes Valiant's class of quasipolynomial-computable families of multivariate

Some Exact Complexity Results for Straight-Line Computations over Semirings

The problem of computing polynomials in certain semmngs is considered and it is shown that the use of branching can exponentially speed up computations using the min, + operations, and that subtraction can exponentiallySpeed up arithmetic computations.

Exponential Lower Bounds for Depth 3 Arithmetic Circuits in Algebras of Functions over Finite Fields

  • D. GrigorievA. Razborov
  • Mathematics, Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 2000
An exponential lower bound on the complexity of depth 3 arithmetic circuits computing some explicit functions f:(F*)n→F (in particular, the determinant of a matrix) is proved.

Completeness classes in algebra

The aim of this paper is to demonstrate that for both algebraic and combinatorial problems this phenomenon exists in a form that is purely algebraic in both of the respects (A) and (B).

Computing Algebraic Formulas Using a Constant Number of Registers

It is shown that, over an arbitrary ring, the functions computed by polynomial-size algebraic formulas are also computed by algebraic straight-line programs that use only three registers, which is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas.