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4
votes
1answer
119 views

Implications of a recent negative result to geometric complexity

A paper was posted in arxiv http://arxiv.org/pdf/1512.03798.pdf titled 'Rectangular Kronecker coefficients and plethysms in geometric complexity theory' by Christian Ikenmeyer and Greta Panova with ...
4
votes
0answers
143 views

What is the status of Determinantal Complexity of Permanent

Recently Landsberg and Ressayre announced an exponential lower bound for permanent's determinantal complexity assuming certain symmetry conditions. What is the status of the problem of Permanent's ...
2
votes
0answers
89 views

Complexity of a particular determinant

Suppose we have an $n\times n$ matrix $A$ with non-negative integer entries such that $\mathsf{Tr}(A^i)=0$ at every $i\in\{1,2,\dots,n-2,n-1\}$ and $\mathsf{Tr}(A^n)\neq0$, then from Trace-Determinant ...
11
votes
1answer
470 views

Expressing Determinant as Permanent

One major problem in TCS is the problem of expressing a permanent as a determinant. I was reading Agrawal's paper Determinant Versus Permanent and in one paragraph he claims the reverse problem is ...
4
votes
1answer
204 views

state-of-the-art bit complexity of the determinant

I'm trying to understand the full bit-complexity of computing the determinant of an $n\times n$ integer matrix, with each entry represented by $M$ bits. I would like to know what is the state-of-the-...
11
votes
2answers
732 views

Determinants and Matrix Multiplication - Similarity and differences in algorithmic complexity and arithmetic circuit size

I am trying to understand the relation between algorithmic complexity and circuit complexity of Determinants and Matrix Multiplication. It is known that the determinant of an $n\times n$ matrix can ...
9
votes
2answers
465 views

Cancellation and determinant

Berkowitz algorithm provides a polynomial size circuit with logarithmic depth for determinant of a square matrix using matrix powers. The algorithm implicitly uses cancellation. Is cancellation ...
16
votes
1answer
431 views

Implications of approximating the determinant

It is known that one can compute exactly the determinant of an $n\times n$ matrix in determinstic $\log^2(n)$ space. What would be the complexity implications of approximating the determinant of a ...
8
votes
0answers
223 views

Grigoriev-Karpinski for the permanent

Grigoriev and Karpinski (ps.Z) showed that any depth-3 circuit over a fixed finite field computing $\mathrm{Det}_n$ requires $2^{\Omega(n)}$ size. I had the misconception(?) until recently that the ...
8
votes
0answers
186 views

Counting small terms in a determinant calculation over polynomials (counting spanning trees by weight)

I have a $n\times n$ matrix $A$. It's terms are $a_{ij}=-x^{w_{ij}}$ if $i\neq j$ and $a_{ii}=\sum_{j=0}^{n+1} x^{w_{ij}}$ on the diagonal. The matrix is symmetric as $w_{ij}=w_{ji}$. Numbers $w_{ij}$ ...
21
votes
2answers
970 views

Lower bound for determinant and permanent

In light of the recent chasm at depth-3 result (which among other things yields a $2^{\sqrt{n}\log{n}}$ depth-3 arithmetic circuit for the $n \times n $ determinant over $\mathbb{C}$), I have the ...
13
votes
1answer
585 views

Smallest known formula for the determinant

The smallest known formula for the determinant has size $n^{\mathcal O(\log n)}$ according to the folklore (or to Ran Raz in its paper Multi-Linear Formulas for Permanent and Determinant are of Super-...
13
votes
2answers
639 views

Gaussian Elimination in terms of Group Action

Gaussian elimination makes determinant of a matrix polynomial-time computable. The reduction of complexity in computing the determinant, which is otherwise sum of exponential terms, is due to ...