Computing permanents over fields of characteristic 3: where and why it becomes difficult

@article{Kogan1996ComputingPO,
  title={Computing permanents over fields of characteristic 3: where and why it becomes difficult},
  author={G. P. Kogan},
  journal={Proceedings of 37th Conference on Foundations of Computer Science},
  year={1996},
  pages={108-114}
}
  • G. Kogan
  • Published 14 October 1996
  • Computer Science
  • Proceedings of 37th Conference on Foundations of Computer Science
In this paper we consider the complexity of computing permanents over fields of characteristic 3. We present a polynomial time algorithm for computing per(A) for a matrix A such that the rank rg(AA/sup T/-I)/spl les/1. On the other hand, we show that existence of a polynomial-time algorithm for computing per(A) for a matrix A such that rg(AA/sup T/-I)/spl ges/2 implies NP=R. As a byproduct we obtain that computing per(A) for a matrix A such that rg(AA/sup T/-I)/spl ges/2 is P(mod3) complete. 

Some facts on Permanents in Finite Characteristics

TLDR
The following paper extends the polynomial-time computability of the permanent over fields of characteristic 3 for k-semi-unitary matrices to study more closely the case k > 1 regarding the (n-k)x-sub-permanents (or permanent-minors) of a unitary nxn-matrix and their possible relations.

Matrix Choosability

TLDR
The permanent lemma of Noga Alon proves that if perm(A)?0, then A has this property for k=1, and a theorem is presented which generalizes both of these facts and is applied to prove “choosability” generalizations of Jaeger's 4-flow and 8-flow theorems in Zkp.

New Hardness Results for the Permanent Using Linear Optics

TLDR
A collection of new results about matrix permanents that are derived primarily via linear optical techniques are presented, which show that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly despite the fact that they can be efficiently sampled by a classical computer.

On the (In)tractability of Computing Normalizing Constants for the Product of Determinantal Point Processes

TLDR
The computational complexity of computing the product of determinantal point processes as a natural, promising generalization of DPPs is studied and the existence of efficient algorithms for this task is ruled out, unless input matrices are forced to have favorable structures.

Computational Complexity of Normalizing Constants for the Product of Determinantal Point Processes

TLDR
The product of determinantal point processes (DPPs), a point process whose probability mass is proportional to the product of principal minors of multiple matrices, is considered as a natural, promising generalization of DPPs and fixed-parameter tractable algorithms are presented.

The Permanent Rank of a Matrix

  • Yang Yu
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1999
TLDR
Several results on perrank of a matrix are proved, motivated in part by the Alon?Jaeger?Tarsi Conjecture, which defines the perrank to be the size of the largest square submatrix of the matrix with nonzero permanent.

Model Theory in Computer Science: My Own Recurrent Themes

TLDR
I review my own experiences in research and the management of science and suggest ways in which science and management can be improved.

Note on the Permanent Rank of a Matrix

De ne the perrank of a matrix A to be the size of a largest square submatrix of A with nonzero permanent. Motivated in part by the Alon-Jaeger-Tarsi Conjecture [3], we prove several results on

References

SHOWING 1-10 OF 13 REFERENCES

The Complexity of Computing the Permanent

  • L. Valiant
  • Mathematics, Computer Science
    Theor. Comput. Sci.
  • 1979

Handling exceptions

Matrix analysis

TLDR
This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.

On computing the permanent in fields of characteristic 3, part I

  • Technion-Israel Institute of Technology,
  • 1996

Basic Algebraic Geometry, volume 213 of Grundlehren

  • 1974

2 Introducing ~(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Computing ~(X), part I

  • 2 Introducing ~(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Computing ~(X), part I

MatrixAnalysis

  • 1985

Permanents, volume 6 of Encyclopedia of Mathematics and its Applications

  • 1978

5 Computing ~(X), part II . . . . . . . . . . . . . . . . . . . . . . . . 23 4.6 Computing ~ 25 5 Proof of the 1{semi{unitary case 27 5.1 Evaluating ~

  • 5 Computing ~(X), part II . . . . . . . . . . . . . . . . . . . . . . . . 23 4.6 Computing ~ 25 5 Proof of the 1{semi{unitary case 27 5.1 Evaluating ~