All Questions
1,808 questions
6
votes
2
answers
1k
views
MIP*=RE theorem and its impact on logic and proof theory
In the monumental paper MIP*=RE five authors, Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, managed to show that two complexity classes: RE and MIP* do in fact coincide. ...
1
vote
0
answers
51
views
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 ...
1
vote
0
answers
67
views
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 ...
1
vote
0
answers
145
views
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}~(\...
1
vote
0
answers
94
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
0
votes
1
answer
114
views
Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
15
votes
1
answer
1k
views
(Very) Large numbers, Chaitin's incompletness theorem and a specific upper bound
Chaitin's incompleteness theorem roughly saying states that for any theory $S$ there exists universal constant $L$ that for any string $\sigma$ one cannot prove (within this theory) that $K(\sigma)>...
2
votes
1
answer
174
views
Optimization over permutation
The Problem
This is the problem I am working on: Given a set $X = \{x_1, x_2, \cdots , x_n\}$ in a metric space, find an optimal ordering $\pi : X \rightarrow X$ that maximizes the following objective ...
1
vote
0
answers
114
views
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 ...
2
votes
1
answer
240
views
Is the problem of vertex enumeration from an H-representation of a polytope NP-hard?
According to the Wikipedia page on the issue, the vertex enumeration problem is NP-hard.
However, double description and reverse linear search are algorithms listed to solve the problem. Moreover, ...
1
vote
0
answers
35
views
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 ...
0
votes
0
answers
61
views
On diagonalizations over complexity classes
I am looking for the following PhD thesis, but could not find it, and all my attempts for finding it failed.
I am wondering if there is a way to get it:
On diagonalizations over complexity classes
By: ...
0
votes
0
answers
55
views
Relationship of optimal solutions between the total function and the sub function
This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
0
votes
0
answers
85
views
Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points
Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$
\begin{align}
\max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
2
votes
1
answer
46
views
Complexity for determining whether a given metric space is hyperconvex?
Suppose I am given a finite metric space as a distance matrix. What is the complexity of determining whether this metric space is hyperconvex?
Definition: A metric space is said to be hyperconvex if ...
1
vote
1
answer
69
views
$A^*$ algorithm to find shortest path when weights in my graph are the inverse of distance
Given a graph G=(V,E) where the weights on my edges are inverse of Euclidian distance between nodes, I want to know if I can use A* algorithm to find the shortest path. How I need to modify the ...
6
votes
0
answers
151
views
Complexity of continued fraction arithmetic operations
Let $A = [a_0; a_1, \dots]$ and $B = [b_0; b_1, \dots]$ be continued fractions.
Let's say that we want to compute $A+B$ or $A \cdot B$ while staying in the continued fraction representation. So, for ...
2
votes
1
answer
125
views
The counterpart of productive set with polynomial computational complexity
For definition of productive set, see here and here, that is defined with computability, or computable function. Restricting computable function as function of polynomial computational complexity, is ...
3
votes
1
answer
807
views
Language equivalence between deterministic and non-deterministic counter net
One-Counter Nets (OCNs) are finite-state machines equipped with an integer counter that
cannot decrease below zero and cannot be explicitly tested for zero.
An OCN $A$ over alphabet $\sum$ accepts a ...
8
votes
1
answer
454
views
Classes of groups with polynomial time isomorphism problem
It is known that the isomorphism problem for finitely presented groups is in general undecidable. What are some classes of groups whose isomorphism problem is known to be solvable in polynomial time? (...
1
vote
1
answer
119
views
Optimization on non-convex set
Let $\Omega$ be an open bounded subset of $\mathbb{R}^2$ and $f\in L^2(\Omega)$ be a given function. Consider the optimization problem
$$\mathrm{min} \int_\Omega u(x) f(x) \,dx\,,$$
where a minimum is ...
-3
votes
1
answer
531
views
Counter net decidability [closed]
Let one Deterministic Counter Net ($\mathrm{1DCN}$), which is a finite-state automata where every state is complete means all states has transition of all input symbols and their respective weight ...
5
votes
0
answers
180
views
Is the matrix multiplication exponent $\omega$ independent from the choice of the base field
The matrix multiplication exponent, usually denoted by $\omega_{F}$, is the smallest real number for which any two $n\times n$ matrices over a field $F$ can be multiplied together using ${\...
10
votes
1
answer
518
views
Can $N!$ be computed in less than $\mathcal{O}(N)$ operations?
The standard algorithm to compute the factorial function $N!$ via repeated multiplications has complexity $\mathcal{O}(N)$, in the model in which each operation costs 1, no matter how many digits the ...
1
vote
0
answers
34
views
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 $NC$ ...
0
votes
2
answers
530
views
Any idea of solving an optimization problem with cubic constraints?
I have the following optimization problem with cubic constraints, which is hard to solve. Are there any ideas, or related references, of solving such a problem?
$$ \begin{array}{ll} \underset {y, z} {\...
1
vote
0
answers
101
views
On determinant and permanent of certain homotopy defined simple matrices
Let $A_1,A_2,B_1,B_2$ be four $n\times n$ $0/1$ square matrices where $$\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$$
hold ($per$ refers to permanent).
I. What ...
1
vote
0
answers
161
views
On an optimization question
Suppose we have a square matrix $M=(1-z)A+zB$ where $A,B$ have integer entries from $\{0,1\}$ with $\det(A)+\det(B)=1$ and $\det(A),\det(B),per(A),per(B)\in\{0,1\}$ and we want to find $z\in[0,1]$ ...
0
votes
0
answers
304
views
A question regarding an unprovability proof
Let LA denote polynomial time arithmetic, Con_LA the equation stating the
consistency of LA, LAJ the system LA+Con_LA, and E2A double exponential time
arithmetic.
A manuscript of mine provides a proof ...
0
votes
0
answers
146
views
Maximizing the norm of a sum of Hermitian matrices
Consider the following problem:
Problem: Given $n\times n$-Hermitian matrices $A_1,\dots,A_r$, find $e_1,\dots,e_r\in\{-1,1\}$ such that $\|e_1A_1+\dots+e_rA_r\|_\infty$ is maximized. Here the norm is ...
1
vote
1
answer
201
views
Diagonally dominant matrix via rows permutation
Diagonally dominant matrices are required in many linear algebra algorithms such as the Gauss-Seidel algorithm.
Some matrices can be made diagonally dominant by permuting its rows and others cannot.
...
1
vote
0
answers
223
views
Fastest algorithm for finding the closest semi-definite matrix?
Given a real-valued, symmetric matrix $A \in \mathbb{R}^{n \times n}$, I'm interested in finding the closest positive semi-definite matrix $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...
1
vote
0
answers
61
views
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 ...
1
vote
0
answers
56
views
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\...
2
votes
0
answers
221
views
Modular inverse computation - avoiding Euclidean algorithm
Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...
1
vote
2
answers
121
views
How to solve the optimization problem $\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$?
I am looking for an algorithm to solve the following optimization problem
$$\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$$
where $\mathbf{w}$ and each $\mathbf{x}_i\in\mathbb{R}^d$.
...
5
votes
0
answers
129
views
Finding an $\mathbb{F}_q$-point on one specific intersection of quadrics
Let $\mathbb{F}_q$ be a finite field of large characteristic and $a_1, a_2, \cdots, a_n \in \mathbb{F}_q$ be some pairwise different elements. I assume that $\sqrt{-1} \in \mathbb{F}_q$. Consider the ...
3
votes
0
answers
105
views
Techniques for solving linear inequalities
For $n$ real variables $x_1, \ldots, x_n$, I have a bunch of inequalities of form $2 x_i > x_j + x_k$ or $2 x_i < x_j + x_k$, where $i,j,k$ are distinct. My goal is to determine whether this set ...
0
votes
1
answer
103
views
Constrained linear optimization problem on $C^1$
I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
1
vote
1
answer
474
views
Is non-convex optimisation really in NP class?
Crossposted on Mathematics SE
I've seen in many optimisation papers the statement that general non-convex optimisation problem is NP-hard. If we assume that non-convex optimisation is in NP class, it ...
1
vote
0
answers
36
views
Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?
Suppose that $P$ is a polyhedron represented by
$$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$
and $P$ contains interior points. Moreover, the ...
2
votes
0
answers
245
views
Pancake sorting problem – Is computing f(n) NP-hard?
The so-called Pancake flipping problem first discussed by Jacob E. Goodman here yields two entangled problems:
MIN-SBPR (Sorting By Prefix Reversals) - Given a permutation, find the smallest sequence ...
3
votes
1
answer
332
views
Complexity of inverting and multiplying against a symmetric Toeplitz matrix with two repeated entries
I know that the computational complexity of inverting a general $n \times n$ matrix $A$ is $O(n^{2.373})$ and multiplying it against an $n \times m$ matrix is $O(n^2m)$. Moreover, I've seen that ...
2
votes
2
answers
235
views
Is it NP-hard to find the min set of nodes in a graph so that the set of paths joining them cover all the nodes?
Given a directed graph $G=(V,A)$ and given for every pair of nodes $(i,j)$
a valid path $P(i,j)=(v_1=i,...,v_l=j)$ on $G$.
Find a minimum set of nodes $M$ such that $\bigcup_{(i,j)\in M\times M}P(i,j)=...
0
votes
0
answers
145
views
Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
0
votes
0
answers
59
views
NC0 randomness vs. non-uniformity
In
Ajtai and Ben-Or. A theorem on probabilistic constant depth
Computations. STOC '84, 1984
Ajtai and Ben-Or show a non-uniform derandomization of BPAC0.
Is there a similar relation known for ...
8
votes
0
answers
360
views
Worst-case complexity of calculating homotopy groups of spheres
Is the best known worst-case running time for calculating the homotopy groups of spheres $\pi_n(S^k)$ bounded by a finite tower of exponentials? How high is a tower? Does $O(2^{2^{2^{2^{n+k}}}})$ ...
0
votes
0
answers
84
views
1-degree SOS proof refutes Linear Programming
I am trying to understand Sums-of-Squares proof systems.
A degree $d$ Sums-of-Squares refutation for a set of polynomial equations $P = \{p_1(x) = 0, ..., p_m(x) = 0\}$ is defined as
$\sum_{i=1}^m g_i(...
1
vote
1
answer
119
views
Adding linear constraint to the domain
I don't know if it is a well-known problem, but I have been struggling to come up with an algorithm.
I have a set of linear constraints $Ax\le b$, $b\ge 0$ ($b$ and $A$ are given, $x$ is a variable). ...
1
vote
0
answers
96
views
On optimizing a multivariate quadratic function subject to certain conditions
The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...