# Determinant versus permanent

@inproceedings{Agrawal2006DeterminantVP, title={Determinant versus permanent}, author={Manindra Agrawal}, year={2006} }

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…

## 13 Citations

### Permanent vs. Determinant

- Mathematics
- 2014

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…

### AN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM

- Mathematics
- 2012

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

- MathematicsSTOC
- 2008

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

- Mathematics, Computer Science
- 2012

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

- MathematicsElectron. 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

- Mathematics
- 2011

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

- MathematicsISAAC
- 2013

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

- Mathematics, Computer Sciencecomputational complexity
- 2012

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).

### The simple, little and slow things count: on parameterized counting complexity

- MathematicsBull. EATCS
- 2015

This contribution is intended to be a self-contained and minimally technical exposition of the material in this thesis, which investigates the complexity of combinatorial counting problems in the frameworks of parameterized (and exponential-time) complexity.

### Sparse instances of hard problems

- Computer Science, Mathematics
- 2011

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.

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