# Questions tagged [permanent]

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### Is $MSB$ of permanent and certifying half number of witnesses easy?

Can there be a $P$ algorithm to decide if number of perfect matchings is at least $(n!/2)+1$ for a bipartite graph on $n+n$ vertices? Can there be a $P$ algorithm to decide if number of witnesses ...
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### Given $n\times n$ matrix $A$ with integer entries, find #$k$SAT formula that yields $\mathrm{perm}(A)>0$

For each #$k$SAT instance one can build a matrix $A$ such that $\mathrm{perm}(A) = F(\Sigma)$, where $\Sigma$ is the solution count of the $k$SAT formula and $F$ an easy to invert function. My ...
101 views

### Computational hardness for sampling a uniform matching

A famous result of Jerrum, Sinclair, and Vigoda shows that there exists a polynomial-time algorithm which takes a bipartite graph $G$ and produces a random perfect matching $M$ of $G$ (assuming one ...
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### Complexity class of approximating perfect match count

We know we can approximate perfect matching count of bipartite and approximate volume of convex bodies in randomized polynomial time. Is there any evidence these approximations could be in Nick's ...
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### Is Permanent $+$-reducible?

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$. Is ...
89 views

### A succinct version of permanent that is $EXP$-complete

Succinct version of permanent is $NEXP$-hard (https://eccc.weizmann.ac.il/report/2012/086/) and so unlikely to be $EXP$-complete. Permanent mod $2$ is in $\oplus L$ and so succinct version is ...
82 views

### On permanent mod $3^t$ computation

By $'$ I mean transpose. I gather the info here from rjlipton.wordpress.com/2014/12/21/modulating-the-permanent. We know that if $U\in\Bbb F_{3^t}^{n\times n}$ satisfies $UU'=I_n$ in $\Bbb F_{3^t}$ ...
70 views

### Approximating max degree $3$ perfect matching count?

We do not have a deterministic constant factor approximation scheme for general $n\times n$ $0/1$ permanent. What is the best factor in deterministic approximation schemes if we only care counting ...
105 views

### What is known about counting bipartite perfect matching with average degree in $[2,3]$ and max degree $3$?

We know counting perfect matching for bipartite graphs with vertex degree $2$ is in $P$ while counting perfect matching for graphs with vertex degree $3$ is in $\#P$. We also know there are degree $3$...
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### Application of weak determinantal identities to GCT?

In general determinants have many identities. Would it help the $GCT$ program by invoking the paradigm of identities such as to state that if the permanent is converted to determinant then it has to ...
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### On permanent of $\{\pm1,0\}$ matrices

Consider the problem of computing the permanent $Per(M)$ of a matrix $M\in\{0,-1,1\}^{n\times n}$ such that the result is bounded in absolute value, $|Per(M)|<B$ where $B$ is part of input. Is ...
169 views

### Complexity of computing generalised determinants. (P - #P transition)

Computing the determinant of a matrix can be done in polynomial time, while computing the permanent is known to be #P-hard. Let $A$ be an $n \times n$ matrix. Define a generalised determinant function ...
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### 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 ...
172 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 ...
127 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 ...
636 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 ...
274 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 ...
295 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?
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### 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 ...
178 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 $r \in [0,2]$. ...
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### 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 ...
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### A question to the #P-complete proof of the permanent from Ben-Dor/Halevi

In the paper of Ben-Dor/Halevi  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 ...
501 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 ...
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### 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 ...
579 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 ...
536 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 ...
247 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: ...
349 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 ...
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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?
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### 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 ...
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 $k$, ...