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1
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0answers
66 views

Quantum algorithms for generalizations of determinants

There are a wide variety of determent-like constructions. Some like the permanent or immanents are variations on the ordinary determinant for matrices over fields or commutative rings. Some like ...
5
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1answer
116 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 ...
5
votes
2answers
382 views

Matrix permanent is 0

Valiant's theorem says that computing the permanent of an $n\times n$ matrix is #P-hard. Is the problem of determining if a permanent is 0 any easier? This arises in the context of sequence A006063 in ...
3
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1answer
125 views

The complexity of decomposing a bi-stochastic matrix

A bistochastic matrix $A$ is a matrix with positive entries in which each row/column sums to $1$. By the Birkhoff von-Neumann theorem $A$ is a convex combination of permutation matrices. Further, by ...
4
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0answers
135 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 ...
8
votes
1answer
170 views

What is largest class of functions $C$ such that we know $\#P$ in not contained in $C$-generated $TC^0$?

In [1] it is stated that "It remains an open question as to whether every function in $\#P$ has $TC^0$ circuits (although it is at least known that not all $\#P$ functions have DLogTime-uniform ...
11
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1answer
467 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 ...
5
votes
1answer
155 views

Size of Formulas with no negative sign for Matrix Permanent

What is the best lower bound for algebraic formulas for Permanent of a matrix given that the formulas have no negative sign? Is there an exponential lower bound known for such formulas and what would ...
3
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0answers
114 views

Computing the permanent with polylog size matrices

The complexity of computing the permanent of a $l\times l$ binary matrix is known to be $\#\mathsf{P}$-complete, from the famous result of Valiant, where $l = \Theta(n)$. We know that the problem is ...
9
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2answers
459 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 ...
8
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0answers
219 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 ...
0
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2answers
260 views

The complexity of computing the permanent of a matrix of zeroes and ones versus a matrix of integers

How much easier is computing the permanent of a matrix with only zeroes and ones than a matrix of only integers?
21
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2answers
963 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 ...
3
votes
0answers
164 views

An ETH-hardness sparsity transition for the permanent

Let $A$ be an $n \times n$ matrix with $0$ or $1$ as entries. Under ETH, the permanent of $A$ cannot be calculated in $exp(o(n))$ time. Consider $A$ has $O(n^{r})$ entries as $0$ where $r \in [0,2]$. ...
3
votes
1answer
219 views

examples of use of permanents

It is known that if calculating permanent is easy, then solving hard problems in NP is easy. Is there a transparent example regarding application of say finding independent set or find chromatic ...
14
votes
2answers
329 views

A question to the #P-complete proof of the permanent from Ben-Dor/Halevi

In the paper of Ben-Dor/Halevi [1] it is given another proof that the permanent is $\#P$-complete. In the later part of the paper, they show the reduction chain \begin{equation} \text{IntPerm} \propto ...
8
votes
1answer
491 views

Conditional results implying difficulty of improving upper/lower bounds for permanent

Let $A$ be a given square matrix. Is there any evidence that beating quadratic lower bounds for $B$ such that $\text{det}(B) = \text{per}(A)$ could be hard? Is there any plausible conjecture which ...
7
votes
1answer
429 views

Exact arithmetic complexity of Ryser's formula for computing permanent

What is the exact number of multiplication operations and addition operations needed to calculate the permanent in Ryser's formula (both original and the Gray coded version)? I am looking reference ...
8
votes
1answer
522 views

Permanent of a $3 \times 3$ and $4 \times 4$ matrix from determinants

Let $A$ be a $3 \times 3$ or a $4 \times 4$ matrix with entries $a_{ij}$. Can someone provide me a matrix $B$ so that $\operatorname{per}(A) = \det(B)$? What is the smallest explicit $B$ that is known ...
5
votes
1answer
484 views

Permanent as projection of determinant and another permanent

I am looking for explicit examples where permanent of a given matrix $A$ is given by a determinant of a larger matrix $B$ (projection in the sense of Valiant). Is there any reference where I can find ...
-2
votes
1answer
238 views

Complexity of counting the number of Good-perfect matching in the bipartite graph

Let's $G=(U, V, E)$ be a balanced bipartite graph which $|U|=|V|=n$ and $|E|=n*(n-1)$; All nodes in $U$ are connected to all nodes in $V$ except $u_i$ to $v_i$ for $1\leq i \leq n$. Definition1: ...
1
vote
1answer
318 views

The Relationship between P^NP and the Permanent

In the lecture notes Introduction to Complexity Theory by Goldreich, there is a section called "How close is $\#P$ is to $NP$". It is stated there that a $P^{NP}$ machine would approximate $\#P$ in ...
20
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5answers
2k views

About properties of adjacency matrix when a graph is planar

1- Is there any specific properties for adjacency matrix when a graph is planar? 2- Is there any thing special for computing the permanent of adjacency matrix when a graph is planar?
23
votes
2answers
716 views

Is there a direct/natural reduction to count non-bipartite perfect matchings using the permanent?

Counting the number of perfect matchings in a bipartite graph is immediately reducible to computing the permanent. Since finding a perfect matching in a non-bipartite graph is in NP, there exists ...
6
votes
1answer
565 views

Permanents - Approximation and connection to integer factorization

Given a non-negative integer matrix of size $n \times n$. If the permanent can be calculated in $n^{2}$ arithmetic operations but each operation is on word size $n^{k}$ bits for some constant $k$, ...
16
votes
1answer
691 views

Can we decide whether a permanent has a unique term?

Suppose we are given an n by n matrix, M, with integer entries. Can we decide in P whether there is a permutation $\sigma$ such that for all permutations $\pi\ne\sigma$ we have $\Pi M_{i\sigma(i)}\ne ...
14
votes
1answer
399 views

Do the proofs that permanent is not in uniform $\mathsf{TC^0}$ relativize?

This is a follow up to this question, and is related to this question of Shiva Kinali. It seems that the proofs in these papers (Allender, Caussinus-McKenzie-Therien-Vollmer, Koiran-Perifel) use ...