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Questions tagged [monotone]

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3
votes
0answers
83 views

Why do most 0/1 matrices need linear arithmetic circuits of size $\Omega(n^2/\log(n))$?

I am reading Alon et al.'s paper Linear Circuits over $GF(2)$ and I am having trouble seeing the counting argument showing that most matrices need a circuit of size $\Omega(n^2/\log n)$. This result ...
1
vote
0answers
36 views

Monotone complexity of PLP

Blum and Nisan show Positive Linear Programming could be done in $NC$ if we only ask for approximate solutions. This paper https://pdfs.semanticscholar.org/8dc7/5aa9d72864022d848c3e599c5f24d9d527e7....
13
votes
2answers
626 views

Counting the number of satisfying assignments in a POSITIVE CNF-SAT

We know the problem of counting the number of satisfying assignment in a given general boolean formula (CNF-SAT), a given DNF formula, or even a given 2SAT formula is a #P-complete problem. Now, ...
14
votes
0answers
280 views

Fourier spectrum of the parity of two monotone Boolean functions

This is a question that I've been pondering, on and off, for a while, and unsuccessfully. I'd be very interested in any insight regarding this conjecture. (Or rather, these conjectures.) Recall that, ...
7
votes
1answer
212 views

Is a monotone boolean function monotone as a multilinear polynomial?

Let $f:\{-1,1\}^n \rightarrow \{0,1\}$ be a monotonically increasing boolean function. That is, $f(x) \leq f(y)$, if coordinate-wise $x \leq y$. Now, let $F: [-1,1]^n \rightarrow \mathbb{R}$ be $f$ ...
6
votes
1answer
205 views

Arithmetic complexity of matrix powering

Assume $M\in\Bbb Z_{\geq0}[x_1,\dots,x_n]^{m\times m}$ be an $m\times m$ matrix in $n$ variables. We know that size of smallest formula that computes $\mathsf{Tr}(M^d)$ where $d\in\Bbb N$ could be ...
14
votes
2answers
636 views

How many negations do we need to compute monotone functions?

Razborov proved that the monotone function matching is not in mP. But can we compute matching using a polynomial size circuit with a few negations? Is there a P/poly circuit with $O(n^\epsilon)$ ...
2
votes
0answers
119 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 ...
13
votes
2answers
368 views

What is an equivalent definition of mP/poly in terms of a Turing machine?

P/poly is the class of decision problems solvable by a family of polynomial-size Boolean circuits. It can alternatively be defined as a polynomial-time Turing machine that receives an advice string ...
45
votes
0answers
1k views

Monotone complexity of s-t connectivity

In the problem CONN, we obtain a directed $n$-vertex graph (encoded as a boolean string of $n^2$ bits, one for each potential edge), and want to decide whether there is a path between all $n^2$ pairs $...
12
votes
1answer
420 views

Monotone circuit complexity of computing functions on sparse inputs

The weight $|x|$ of a binary string $x\in\{0,1\}^n$ is the number of ones in the string. What happens if we are interested in computing a monotone function on inputs with few ones? We know that ...
22
votes
4answers
1k views

Monotone arithmetic circuits

The state of our knowledge about general arithmetic circuits seems to be similar to the state of our knowledge about Boolean circuits, i.e. we don't have good lower-bounds. On the other hand we have ...
14
votes
1answer
395 views

Can you decide equivalence for monotone Boolean expressions that do not contain negation in PTIME?

Is the following problem in PTIME, or coNP-hard: Given two Boolean expressions $e_1$ and $e_2$ in variables $x_1,\dots,x_n$, without negation (ie, the expressions are entirely built up via $\wedge$ ...