Questions tagged [computational-complexity]
This is a branch that includes: computational complexity theory; complexity classes, NP-completeness and other completeness concepts; oracle analogues of complexity classes; complexity-theoretic computational models; regular languages; context-free languages; Komolgorov Complexity and so on.
1,300
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Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?
Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ are irreducible and of equal degree what is the best ...
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82
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Complexity of quadratic polynomials isomorphism
Two polynomials $f,g$ are isomorphic iff $f(x_1,\ldots x_n)=g(\pi(x_1, \ldots x_n))$ for a permutation $\pi$.
$f,g$ are equivalent if there exists invertible linear transormation
$A$ such that $f(X)=...
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354
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Analysis of a partition algorithm
EDIT:
I realized that if $J$ is not a solution so is $J^c$. I updated the algorithm because of this.
Given some positive integers $x_1,\cdots, x_n$.
The following algorithm is for solving the ...
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760
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What is the fastest way to sort numbers lexicographically?
I have $N$ sequences of numbers. None of them is longer than $10^6$. I want to sort those sequences lexicographically. For example, given sequences {1, 2, 4}, {1, 2, 3}, {2, 5, 7}, {2}, I want to have ...
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2
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252
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Maximum subgraph edge distance greater than given number
I have a weighted graph G with approximately 75000 nodes. I would like to find subgraph G' induced on a subset of nodes, such that all edge weights in G' are greater than a given constant C and the ...
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224
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About expectation norms on graphs
Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= \frac{E(S,\bar{...
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177
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A particular argument in the review on expanders by Hoory-Linial-Wigderson
I am thinking about the third bullet point on page 455 here, http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/
Can someone explain what is the argument there which seems to conclude ...
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113
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Implications of the impossibility of efficient sampling from random non-Hamiltonian graphs
Nisan's answer to this question shows the Impossibility of efficient sampling from random non-Hamiltonian graphs (unless $NP=coNP$). I am interested in the implications of this conjecture.
Does the ...
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1
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303
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Cycles of Permutation Related to Rectangular Matrix Transposition
let the entries of a rectangular matrix $A\in\mathbb{C}^{m\times n}; m,n\in\mathbb{N}$ be stored in row-order in a linear vector $v$, i.e. $A_{i,j}=v_{i*m+j}$
Question:
How can the first element ...
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2
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375
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When is a power of an indeterminate in an ideal with 2 generators?
If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
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895
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How to prove the NP-hardness of this set covering problem
In the Set Covering problem, we are given a ground set $U$ and a collection $S$ of subsets of $U$, where each subset is associated with a non-negative cost, the Set Cover problem asks to find a ...
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225
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Construction of an integral point set given the set of distances, its minimal description to get a measure of its complexity and its unique identifier
Given a set of distances between every pair of points of an integral point set $P$ of $n$ points; say $D = \{{d_i}\}$.
Q1. What is the least time complexity
possible/known for recreating the
...
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1k
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Longest run of heads
Consider $n$ independent tosses of a fair coin; the sample space has $2^n$ elements. Let $R_n(x)$ be the length of the longest run of heads in outcome $x$. We know that $$E[R_n]=\Theta (\log n)$$
csun....
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281
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Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?
I failed to get an answer at https://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here.
In https://math....
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k-uniform k-partite hypergraph matching in polynomial time
I have what seems like an elementary question, but google didn't throw up any answers for it. I would appreciate any pointers that MO users may provide.
It is well known that for $k\geq 3$ finding ...
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256
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Pseudorandom Generator vs Constant Depth Circuits / Branching Programs
Hi.
I am looking for a survey on the state of the art in pseudorandom generators vs
(1) constant depth circuits
and/or
(2) Branching Programs
For (1), is "Anindya De, Omid Etesami, Luca Trevisan ...
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273
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Proof that any NP problem can be reduced (in P time) to any problem in NPC?
Given the seemingly broad definition of NP, it is very interesting that one can prove that any member of NP can be reduced in polynomial time to any member of NPC. (I guess this is true by definition ...
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6k
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Finding the square root modulo n, when the factors of n are known
Last month, I asked whether there is an efficient algorithm for finding the square root modulo a prime power here: Is there an efficient algorithm for finding a square root modulo a prime power?
Now, ...
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0
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81
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Quasi polynomial algorithm for NP complete problem [closed]
I know that quasi polynomial algorithm is neither polynomial nor exponential. But I want to know if we find such algorithm for NP complete problem, will it be of any use? Or is there such algorithm ...
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0
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84
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Obstacles to computing $\pi(n)$ in $O(n^{2/3-\epsilon})$ time
Edit: Apologies, as mentioned in the comments I failed to notice the analytic algorithms that take $O(n^{1/2+\epsilon})$ time, so this question doesn’t make much sense. It’s possible there is a ...
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62
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Primality testing by reversible computation using the prime number theorem
Suppose we want to build a primality testing algorithm for the numbers limited to the set $A =\{1, ..., 2^n\}$ and $n$ is reasonably large. The prime-number theorem tells us that there are ...
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0
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113
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Integration in polynomial time
The work of Friedman and Ko and
Müller guarantee the polynomial time computability of the integrals of analytic functions inside the circle of convergence. But do algorithms have practical value? Is ...
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0
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161
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Fast algorithm for computing certain signal transformations
Let $f,g,h:\mathbb Z\to\mathbb C$ supported on $[-n,n]$. For $\tau\in \mathbb Z$, let $\operatorname{sh}_\tau f$ be the shift of $f$ by $\tau$ (i.e. $(\operatorname{sh}_\tau f)(t) = f(t-\tau)$). ...
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48
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Computing geodesic length of Euclidean lines in the manifold of positive definite matrices
I am working with the manifold of positive definite matrices $PD(n)$ equipped with the affine-invariant Riemannian metric (AIRM) $g_P(V,W):=tr(P^{-1}VP^{-1}W)$, where $P \in PD(n)$ and $V,W \in T_P PD(...
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0
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75
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Circulant matrix inverse in $GF(p)$
For a polynomial $C(x)=c_0+\dots+c_n x^n$, consider a circulant matrix $C$ such that
$$
C= \begin{pmatrix}
c_0 & c_{n-1} & \cdots & c_2 & c_1 \\
c_1 & c_0 &...
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0
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42
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Complexity of the TSP for hypercube graphs
Question:
what is known about the complexity of finding the Hamilton cycle of minimum weight in graphs that resemble hypercubes with weighted edges?
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111
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Sudden drop in complexity class due to the more general correlations
Recently I was asking about the impact of the groundbreaking result MIP*=RE on logic and proof theory (see this discussion). Surprising as it is I got confused with the following: MIP* is a ,,quantum''...
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0
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47
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Hardness of an optimization problem when some variables are fixed
Given a general optimization problem, I would like to know what we can say about the hardness of the problem when a subset of its variables are fixed.
With the two (related) examples, it is clear that ...
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0
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57
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Are the lower elementary functions closed under limited recursion?
The lower elementary functions (also called Skolem elementary functions) are functions generated from the successor, modified subtraction, projection functions by the operations of composition and ...
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0
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135
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Complexity of calculating the expectation of $\operatorname{Tr} h(A)$, $A$ is a random matrix
$A$ is a $d_1\times d_1$ random matrix. Given $\{g_i\}~(1\leq i\leq n)$ iid Gaussian variables, $f_{ij}(g_1,g_2,...,g_n)~(1\leq i,j\leq d_1)$ are degree-$d_2$ polynomials. And $f_{ij}\equiv f_{ji}~(\...
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0
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105
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Computing sine of gamma function [closed]
In the sense of bit complexity, how difficult is it to compute $$\sin(a\Gamma(x))$$ where $a$ is a constant and $x>1$? Is it possible to avoid the computation of $\Gamma$ as first step?
Is there a ...
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0
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28
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Are there any known lower complexity bounds on solving positive semidefinite or positive semidefinite feasibility problems?
I've been trying to attack the problem posted here, about quickly checking if a matrix has any positive semidefinite completions. I suspect that the answer to the question is "no", because ...
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0
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30
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Does $2$ variable linear Diophantine equation in $NC$ imply $2$ dimensional shortest vector is in $NC$?
Consider the Linear Diophantine in known $a,b,c\in\mathbb Z$
$$ax+by=c.$$
Above can be solve by Extended Euclidean which is not in $NC$ as far as we know. It is clear if Extended Euclidean is in $NC$ ...
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0
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101
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On determinant and permanent of certain homotopy defined simple matrices
Let $A_1,A_2,B_1,B_2$ be four $n\times n$ $0/1$ square matrices where $$\det(A_1)=\det(A_2)=per(A_1)=per(A_2)=1$$
$$\det(B_1)=\det(B_2)=per(B_1)=per(B_2)=0$$
hold ($per$ refers to permanent).
I. What ...
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0
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161
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On an optimization question
Suppose we have a square matrix $M=(1-z)A+zB$ where $A,B$ have integer entries from $\{0,1\}$ with $\det(A)+\det(B)=1$ and $\det(A),\det(B),per(A),per(B)\in\{0,1\}$ and we want to find $z\in[0,1]$ ...
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0
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153
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Fastest algorithm for finding the closest semi-definite matrix?
Given a real-valued, symmetric matrix $A \in \mathbb{R}^{n \times n}$, I'm interested in finding the closest positive semi-definite matrix $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...
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0
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59
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Is there any lower bound for basis computation in finite Abelian groups?
Victor Shoup in this paper has given a lower bound for discrete logarithm. The algorithms that I have come across use discrete logarithms (extended discrete logarithms) to compute a basis for a finite ...
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0
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53
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Over a given finite field, how many couples of matrices there are, for which their minimal polynomials are co-prime?
Let ${\mathbb F}_{q}$ be a given finite field. How many couples of $n\times n$ matrices $\left(A,B\right)$ over ${\mathbb F}_{q}$, such that $\gcd\left(\mu_{A}\left(\lambda\right),\mu_{B}\left(\lambda\...
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0
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Is this factorization problem in EXP?
Factorization is not known to have a polynomial time algorithm. Traditionally the input length is number of bits in representation of the integer to be factored.
However now consider integers of form $...
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0
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43
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Computational hardness of a discrete generalized rectangle packing problem
I have a decision problem that is clearly in NP, but I cannot seem to prove that it is in P, nor can I prove its NP-hardness. I attribute this more to my inexperience than to the problem's difficulty (...
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0
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117
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Frog game on tree graphs is in NP but not in P (NP-complete)?
Problem
We can restrict ourselves to tree graphs. What is the complexity of the following problem?
Let $G$ be simple connected graph with vertices in $V$, edges in $E$, and a vertex weighted function $...
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0
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106
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Finding the optimal arithmetic circuit for evaluating a given polynomial
The Horner's algorithm takes as input a univariate polynomial $f(X)$ and an evaluation point $x$ and computes $f(x)$ using $O(\deg(f))$ field operations.
Suppose now that the polynomial $f(X)$ is ...
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0
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69
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What is the complexity of elgamal cryptosystem? [closed]
Its clear generation of keys based
On cyclic group and its generator for z_p
So my question
Does finding the generator efect on complexity
Moreove does the size of message M effect on the complexity?
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41
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If statement in the algebraic group model (AGM)
In the algebraic group model (https://eprint.iacr.org/2017/620.pdf), can one use "if" statement? For example, can one do the following in AGM?
input: x, y, z
if (x = y) then z = x else z = ...
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0
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92
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Formalizing intuition of search hardness
Basically, this is a search problem of an object that is promised to exist. Suppose we have an object that can be described completely and uniquely by $m$ properties (each n bits). Suppose a search ...
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0
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40
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What is the complexity of the matrix multiplication closure for a given generating system?
Given a generating set of $k$ matrices $X = \{M_1, M_2, \ldots, M_k\}$, with $M_i\in \mathrm{Mat}(\mathbb{C},n)$, what is the worst case complexity for computing the algebraic closure w.r.t. matrix ...
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0
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91
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Fast algorithm to compute nimber product
It is known that nimbers (Grundy numbers) below $2^{2^n}$ form a field with the nim addition $\oplus$ and the nim product $\cdot$.
Generally, one can develop an algorithm to compute the product of two ...
1
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0
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66
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On perfect matchings on planar graphs - is there a linear time deterministic algorithm?
The slides here provide a way to get a pfaffian orientation from Minimum Spanning Tree.
MST can be found in linear time if graph is planar and weights are $1$ and the slides give a linear time ...
1
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0
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158
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Does fraction-free Gaussian elimination use fractional row operations?
I would like to understand whether Gaussian elimination of an integer matrix, which uses only row operations of the form
Addition (or subtraction) of row $i$ to row $j$
can be performed in ...
1
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0
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80
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Literature about graph isomorphism and incidence matrix [closed]
I would like to read some paper, if any, for some classes of graphs, regarding inverting (right/left inverting) the incidence matrix to solve the graph isomorphism problem. Or anyway some known facts ...