Questions tagged [computational-complexity]
Use for questions about the efficiency of a specific algorithm (the amount of resources, such as running time or memory, that it requires) or for questions about the efficiency of any algorithm solving a given problem.
3,578 questions
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Find a Context Free Grammar (CFG) with 3p zeroes and 5q ones
How can I construct a context-free grammar for the following language?
The language is: $L = \{{(0 + 1)^* | \text{#}0 = 3p, \text{#}1 = 5q, p, q \geq 0} \}$.
I can construct a CFG for L if the zeroes ...
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Complexity of counting perfect matchings in 3-uniform Hypergraphs (X3C)
I am aware that counting perfect matchings in graphs is #P-complete. I want to know what the complexity is in 3-uniform hypergraphs, i.e., X3C (and more generally in hypergraphs). You could also ...
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What is the fastest known algorithm for evaluating a homogenous binary polynomial?
Let $f(x,y) = \sum_{i = 0}^d f_i x^i y^{d-i}$ be a homogenous binary polynomial of degree $d \in \mathbb{N}$ over a field $k$. I want to evaluate $f$ at a point $P_0 = (x_0, y_0) \in k^2$. What is the ...
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Whether we can prove Error Reduction for $\textbf{RP}$ like $\textbf{BPP}$ or not?
I'm doing exercise 7.4 in Computational Complexity: A Modern Approach.
(Error Reduction for $\textbf{RP}$) Let $L\subseteq \{0, 1\}^∗$ be such that there exists a polynomial-time PTM $M$ satisfying ...
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Is Monotone QSAT PSPACE Complete?
I've seen that "Monotone" is given a few different definitions w.r.t. SAT, but I'm interested in the more common definition: The SAT problem, where each clause contains only unnegated ...
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Hamiltonian path is in randomized polynomial time(RP) complexity
Followed from this question.
Let's define the $\texttt{MinHalfSimpCycle}$ search problem: Given $G=(V, E)$ a complete, undirected graph with weights on the edges. We want a simple cycle in $G$ (each ...
user1421746
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The time complexity of $yx^\top$ where $x,y \in \mathbb R^n$ are non-sparse vectors? [closed]
I want to know the computational complexity of the multiplication $yx^\top$ where $x,y \in \mathbb R^n$ are non-sparse vectors, i.e., this is an outer product. What is also the computational ...
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Formulas for fitting points to a quadratic and log linear equation
I have a set of points $(x,y)$ in a coordinate plane. I understand that quadratic regression is the problem of fitting a parabola of the form:
$$ax^2 + bx + c$$
to those points. Assuming this link is ...
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Minimizing the infinity norm of a matrix
Let $v_1,\dots v_k$ be an orthonormal basis of $\mathbb R^k$ and $d\ll k$. We want to find vectors $u_1,\dots u_d\subset \{v_1,\dots v_k\}$ in order to minimize
$$\left\|\sum_{i=1}^d u_i u_i^\top\...
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Hamiltonian path is $\mathbf{NP}$-complete [closed]
We will define $$\texttt{Hamiltonian}_{\texttt{st}}=\left\{\langle G, s, t\rangle: \text{There is a simple path from $s$ to $t$ passing through all vertices in $G$}\right\}.$$ Here $G$ is an ...
user1421746
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Decomposing the matrix multiplication tensor with unitary 2-designs
The question concerns equation (5) from the paper Designing Strassen’s Algorithm.
The tensor associated with matrix multiplication (say in $(\mathbb{C}^{n\times n})^{\otimes3}$) is given in the paper ...
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Can Floor and Ceiling Be Ignored to Simplify $T(n) = 3T(⌊n/5⌋) + 3T(⌈n/5⌉) + 200n \lg n$ for Master Theorem?
I'm working on a problem set where I need to apply the Master Theorem to several recurrences. One of them is:
$$T(n) = 3T(⌊n/5⌋) + 3T(⌈n/5⌉) + 200n \lg n$$
The question asks to use the Master Theorem ...
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Kolmogorov complexity bound on the prime numbers
This is an advancement of a proof concerning prime numbers, as discussed in the short introduction to Kolmogorov complexity. Unfortunately it has an error somewhere I cannnot see, as it contradicts ...
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Polynomial block with Turing machine [closed]
A function $f$ is said to be thinking in an encapsulated linear memory if:
$f(x)$ is a polynomial block (that is, there exists a polynomial $q$ so that $|f(x)|\leq q(|x|)$) for each input $x$.
...
user1408191
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Is the language $𝐿$ is in $NP\setminus P?$
Suppose that there exists a function $𝑓:\{0,1\}^𝑛 → \{0,1\}^𝑛$
such that, 𝑓 is computable in polynomial time; and the following task cannot be computed in polynomial time (that is, there are $𝑥 ∈ ...
user1408191
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Boolean function recognition by circuit family
We consider Boolean circuits as we do, Specifically, inner nodes are either AND, OR (both – fan-in 2), or NOT (fan-in 1) gates. The fan-out of each gate is 2. The size of a circuit is the number of ...
user1326505
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Circuit complexity in boolean functions
we consider Boolean circuits as we do, Specifically, inner
nodes are either AND, OR (both – fan-in 2), or NOT (fan-in 1) gates. The fan-out of each gate is 2. The size of a circuit is the number of ...
user1326505
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Could the Relativity of Time Impact the P vs NP Problem? [closed]
Given that time behaves differently under relativistic conditions (such as near black holes or in high-speed motion), could the relative nature of time influence the complexity classes P and NP? For ...
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How can the computable numbers be countable?
So Turing proved that computable numbers are countably infinite. I think his reasoning is essentially because you can make a list of Turing machines where each machine appears only once and each ...
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Does it have to be $P = NP$ or $P \neq NP$?
(I know nothing about this topic beyond the popular level, so apologies if this question is not well-posed.)
Every time I've seen this problem discussed, it's always implicit that P must either equal ...
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Identifying whether certain palindromic languages are non-regular [duplicate]
In a nationwide entrance exam conducted in India, the following problem was asked in the year 2007:
Which of the following languages are regular?
(A) $L_1 = \left\{ ww^R \mid w \in \{0, 1\}^+ \right\}$...
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If SubsetSum can be solved in pseudopolynomial time, can we karp reduce SAT/3SAT to it and solve in pseudopolynomial time?
When learning pseudopolynomial time and strong NP-completeness,
this timestamp at this video says:
Which basically
1.SubsetSum can be solved in pseudopolynomial time.
2.SAT, even using unary encoding,...
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Time complexity of solving systems of equations with a sparse structure.
I have a following systems of equations with $A \in \mathbb{R}^{n\times n}, B \in \mathbb{R}^{n\times m}, c\in \mathbb{R}^{m}, a \in \mathbb{R}^{n}$, and $b \in \mathbb{R}^{m}$.
$$
\begin{bmatrix}
A &...
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Extending Quicksort's $\mathcal{O}(n \lg n)$ Bound to Duplicated Elements
This is the final part of Problem 7-2 of CLRS' Introduction to Algorithms. The exercise asks to modify the argument given in the text so that the $\mathcal{O}(n \lg n)$ bound also applies to arrays $A$...
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Question about the example correctness of polynomial time reducibility
in here, when the answer tries to answer about the property of
polynomial time reduction, in the last paragraph it writes:
However, Polynomial-time reductions are not symmetric. That is, it is not ...
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