All Questions
598 questions with no upvoted or accepted answers
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Minimizing quadratic form over permutations
Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem:
$\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$,
where $S_n$ ...
- 1,839
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1k
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Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
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57
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Complexity of evaluation of analytic functions
Given an analytic function $f(x)$ (say as combination of elementary functions and operators), is it possible to compute $n$ first bits of the value of the function on the whole interval $[a, b]$ ...
- 67
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21
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Easy instance of set cover
I am trying to prove that a natural greedy algorithm solves the following instance of the set cover problem: for a set of elements $e\in U$ with a set of weights $w_e$, we define the cost of a subset ...
- 4,049
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75
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Klondike Solitaire as an NP-complete game
I am not a mathematician. I am trying to understand if the paper "The complexity of solitaire" that shows this game is NP-complete also has a implicit assumption that a given hand can only ...
- 1
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37
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Does the same hardness with access to different oracles imply equivalent oracles?
If $\textrm{coNP}^{\textrm{NP}}=\textrm{coNP}^{L}$ for $L\in\textrm{NP}$, then does that make $L$ NP-complete?
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60
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Understanding "A Note on a Theorem by Ladner"
I'm reading A Note on a Theorem by Ladner by Balcazar and Diaz, trying to understand the proof of Theorem 2.1. I don't know if it's just old notation, but I'm having quite a bit of trouble.
In step (...
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122
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Is it in theory possible to create a subexponential algorithm for solving discrete logarithms in multiplicative subgroups or within an Integer range?
As far I understand, when it comes to finite fields, Pollard rho and Pollard’s lambda are still the best algorithm for solving discrete logarithms in a multiplicative subgroup/suborder…
Index calculus ...
- 87
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48
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A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,224
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115
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Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
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92
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Algorithm that can solve or approximate the solution to a combination problem
I have a computational problem on my hands and I would like your help.
Here is my problem (simplified)
Let $X = \{x_1, x_2, \ldots, x_n\}$ represent a set of $n$ values.
Each value $x_i$ has a ...
- 1
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39
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Prove the NP-hardness of the following problem: Whether there exists a partition for a set of data points
Can anybody help me prove the NP-hardness of the following question:
Given $x_0, x_1, ..., x_m \in \mathbb{R}^n$, determine whether there exists a partition $S\cup [m]\backslash S$, such that $x_0 \in ...
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91
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Some new questions on Rademacher complexity
For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable.
Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...
- 185
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39
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Max-flow modeling with unified vehicle and commodity variables
I am working on a network flow problem that involves routing through a time-space network. The network consists of:
A single source node and a single demand node.
A fleet of vehicles with specified ...
- 101
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Application of greedy approach for optimization
I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$
where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
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36
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ILPs with square constraint matrix
Given the Integer Linear Programming ($\text{ILP}$) problem
\begin{array}{ll}
\text{minimize} & c^T x \\
\text{subject to}& \mathbf{A}^T x \ge b \\
\text{where}&c,x,b\in\mathbb{N}_0^n,\\ &...
- 13.2k
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16
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Complexity of finding single source paths with capacity constraints and length constraints
Let $G=(V,A)$ be a directed graph with distinguished vertex $s\in V$ and let $c:A\rightarrow{\mathbb N}$ denote arc capacities. For any $t\in V,t\not=s$ we are given two numbers: $C_{t},L_{t}$. Let $...
- 21
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Monotony of enforced subtour merging
Is it true that for a symmetric TSP instance in the sequence of edges generated by successively:
calculating the optimal 2-factor
adding cardinality constraints on the edgesets of the 2-factor's ...
- 13.2k
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171
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Solve NP-hard type problems with linear programming
I would like to know if there is any way to solve an NP-hard type problem, for example, the TSP, sum of subsets or knapsack problem, by using linear programming and not by brute force.
I ask this ...
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64
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Alternatives to McCormick Envelope
I have an optimization problem for which I have the optimal solution obtained by the ILP.
However, when I introduced the McCormick Envelope to replace the product of a bi-linear term in its LP ...
- 3
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67
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Counting distinct elements in smallest number of queries
There is an array of objects $a_1, \dots, a_n$. For any two objects, we can ask if they're equal or not. Our goal is to find the number of distinct objects in the array by only asking such queries. ...
- 1,171
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164
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Inf-convolution of norm 1 and norm 2 square
The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is
$$
h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) .
$$
We can prove that if $f,g$ are convex functions, then $h$ is convex.
...
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1
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114
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Mixed integer program and continuous Diophantine approximation
Let $n\in\mathbb{N}$ such that $n\geq 2$ and let $0<r<1$ be a real number. We wish to solve the following problem.
$$\min_{(t,(z_j)_{j=2}^n) \in \mathbb{R}\times \mathbb{Z}^{n-1}} t$$
subject to ...
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0
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62
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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: ...
- 32.1k
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55
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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}^...
- 1
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85
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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} [ ...
- 671
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0
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304
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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 ...
- 131
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146
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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 ...
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145
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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\|_{...
- 439
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59
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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 ...
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0
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84
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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(...
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0
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122
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Will an integer program to deterministically factor integers help derandomize $\mathbb F_q[x]$ factoring?
There are many analogies between the objects $\mathbb F_q[x]$ and $\mathbb Z$.
Supposing there is a fixed (say $10^9$) dimension linear integer program (describable without any objective function) in ...
- 13.9k
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272
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Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
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94
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Boolean operation on n dimensional polyhedron
A polyhedron in $R^n$ is defined by a set of half-planes: $P = \{x \in R^n \mid Ax - b \le 0\}$.
Given a set of polyhedra in $R^n$, $ P_1, P_2, \dotsc, P_k$, is there an algorithm/implementation that ...
- 21
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0
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185
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Future of complexity classes in case NP=P
The P=NP question is still unresolved and there is no hope that the situation will ever change.
Assume now the hypothetic situation that P=NP had been confirmed:
Questions:
what will become of the ...
- 13.2k
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119
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Polynomial-time algorithm for exact projection to polyhedral cone
Given $c \in \mathbb{R}^d$ and $A \in \mathbb{R}^{n \times d}$, project $c$ to the polyhedral cone $\{x \in \mathbb{R}^d \mid A x \leq 0\}$. Is there an algorithm that outputs an exact solution to ...
- 2,203
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253
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What is the weakest subsystem of Second-order Arithmetic (or its first-order part) that proves Szemerédi's Regularity Lemma?
The question is in the title. Szemerédi's Regularity Lemma is the following (according to the Wikipedia entry):
For every $\epsilon \gt 0$ and positive integer $m$ there exists an integer $M$ such ...
- 6,132
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84
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Shattering of a set of binary classifiers
Let $S$ be a set, and let $\mathcal{F}_{S}=\{f:S\to\{-1,+1\}\}$ be a set of different label assignments. Show that $\mathcal{F}_{S}$ shatters at least $|\mathcal{F}_{S}|$ subsets of $S$.
Here is what ...
- 143
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1
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397
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What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
- 1
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0
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115
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Explicit equation for border of the Minkowski sum of sets
Assume we have sets of the form
$$
M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}
$$
where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$.
Goal
I am looking for an (explicit) representation ...
- 385
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124
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The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 287
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84
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Decidability of choosing delay in Takens' theorem
In Dynamical systems theory, Takens' embedding theorem is as follows:
Suppose that a measured time series $y(1), y(2), \ldots, y(N)$ lies on a $D$-dimensional attractor of an $n$th-order ...
- 183
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109
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How to find a set given its support function
Let $\mathcal{U}$ be a convex and compact set. Its support function is defined as $\delta^*(v|\mathcal{U})=\sup_{u\in \mathcal{U}} v^T u$. Assume that we are given the support function $\delta^*(v|\...
- 53
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40
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Subtour-gluing constraints for ILP formulation of TSPs
If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-...
- 13.2k
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96
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Why is Gaussian distribution always chosen for smoothed analysis?
I came across the algorithmic perfomance analysis model of smoothed analysis. In all references that I read a Gaussian distribution was used for perturbation (e.g. Spielman and Teng 2004 for the ...
- 29
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116
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Exact Gaussian elimination of a rational matrix
If a matrix $A$ consists of rational elements, and we have access to only row operations of the form
Row addition/subtraction from row $i$ to row $j$
Row exchanging row $i$ with row $j$
What is the ...
- 111
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0
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84
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3-SAT family with $\omega(n^2)$ time complexity
A 3-SAT family is an algorithm that given a positive integer $n$ outputs a 3-SAT problem in $n$ clauses in $O(n^{1+\epsilon})$ time ($\epsilon$ is to allow for the indexing of the variables).
A ...
user409739
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0
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135
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On a deterministic primes search problem
I feel the following problem might be resolved already. But I could not find any related answers.
If $p_1,p_2,\dots,p_t$ are primes where $2\leq t=o(\log n)$ is there a prime within $$\prod_{i=1}^...
- 13.9k
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0
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165
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Minimum circumscribed ellipsoid of $\mathcal H$-polytope
Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I ...
- 235
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0
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104
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Efficient Algorithm to Find Subset of Vectors Over $\mathbb{F}_q$ Living in Low Dimensional Subspace
Let $q$ be a fixed prime, $P, Q$ be polynomials with $\mathrm{deg}(Q) < \mathrm{deg}(P)$ and $h = O(\log n)$.
Let $S$ be a subset of $\mathbb{F}_q^n$ of size $P(n)$ such that there exists a subset ...
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