# 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} }

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.

## 9 Citations

### Some facts on Permanents in Finite Characteristics

- MathematicsArXiv
- 2017

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

- MathematicsJ. Comb. Theory, Ser. A
- 2000

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.

### On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

- Computer Science, MathematicsDiscret. Appl. Math.
- 2001

### New Hardness Results for the Permanent Using Linear Optics

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2016

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

- Computer ScienceICML
- 2020

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

- Computer Science, Mathematics
- 2021

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

- MathematicsJ. Comb. Theory, Ser. A
- 1999

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

- PhilosophyCSL
- 2011

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

- Mathematics
- 1998

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…

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