All Questions
1,809 questions
29
votes
2
answers
1k
views
A combination of two well-known complexity problems
Suppose you are given two graphs $G$ and $H$ and are told that one of the following two situations occurs. Either they are isomorphic, or one of the graphs contains a Hamilton cycle and the other ...
- 29k
1
vote
2
answers
1k
views
Nonstandard Hessian approximations in Gauss-Newton
The Gauss-Newton algorithm optimizes functions
$$
E(x) = \sum f(x)^2
$$
by approximating f as (locally) linear, in which case the Hessian of $E$ is approximated as
$$
H = 2 \sum {J_f}^T J_f
$$
Now ...
- 561
1
vote
2
answers
444
views
Levenberg-Marquadt near the minima for non-zero-residual problems
I'm using the LM algorithm to do gradient descent in a model fitting context. I'm minimizing:
$$
c(x) = \sum ( f_i(x) - y_i )^2
$$
I'm noticing that after a few steps when I'm close to the minima, I ...
- 561
2
votes
1
answer
126
views
Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?
I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise.
The other variables in the linear program,...
- 121
0
votes
0
answers
103
views
Gauss-Newton for quotient functions
I'm optimizing a function of the form
$$
\sum \frac{ \|\mathbf{f_i}(x)\|^2 }{ g_i(x)^2 + h_i(x)^2 }
$$
where $x$ is a real vector, $\mathbf{f}(x)$ is a real vector, and $g(x)$ is a scalar. My first ...
- 561
2
votes
0
answers
179
views
Randomized alternative to Buchberger's algorithm
Richard Lipton's blog describes a A New Way To Solve Linear Equations by Prasad Raghavendra.
Can the ideas in this algorithm be generalized to systems of polynomial equations to provide a randomized ...
- 529
0
votes
1
answer
377
views
Algorithm for vector space
I have $n$ vectors $e_1 \in (\mathbb Z/2 \mathbb Z)^m,\dots,e_n \in (\mathbb Z/2 \mathbb Z)^m $
and a vector $ v \in (\mathbb Z/2 \mathbb Z)^m $
I need to find the better algorithm which answers ...
3
votes
0
answers
445
views
Gaussian Elimination in terms of Group Action
Gaussian elimination makes determinant of a matrix polynomial-time computable. The reduction of complexity in computing the determinant, which is otherwise sum of exponential terms, is due to presence ...
- 245
1
vote
1
answer
769
views
Results for minimizing the norm w.r.t a unitary matrix
Suppose $x \in \mathbb{R}^n$, $B,U \in \mathbb{R}^n\times\mathbb{R}^n$ and $U$ a unitary matrix. Define $g_{U}(x) = || BUx||$ where $||.||$ is some norm or norm-ish function on $\mathbb{R}^n$ (not ...
- 322
4
votes
1
answer
5k
views
Generation of All Path in a Directed Acyclic Graph
I am working on a very large dataset of a single DAG whose vertices have a low branching factor. I need to generate all possible (simple) paths starting from the source and write them to a file.
My ...
- 173
0
votes
0
answers
555
views
VC dimension and boolean hypercube subgraphs
Are there any well studied graph theoretic properties that are common to all subgraphs of the boolean hypercubes that have a given VC dimension d.
- 11
17
votes
2
answers
4k
views
Optimizing the condition number
Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns $...
- 96.4k
7
votes
2
answers
814
views
Building a Physical Model to Solve Sudoku
Before asking my questions, allow me to begin with a separate example to help clarify what I'm driving at. For terms that are not defined formally, please interpret them as you feel would be most ...
- 7,831
0
votes
0
answers
194
views
A linear program related question
Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.
Let $\alpha^k \in (\alpha_1^k, \alpha_2^...
- 769
0
votes
0
answers
79
views
Computing maximum point for minimal function of a family of linear functions
Let $x \in S^n $ where $S^n = ${$ [x_1,x_2,...,x_{n+1}]\in \mathbb{R}^{n+1} \mid x \ge 0 , \sum x_i = 1 $} and let $f_i : I^n \to \mathbb{R}$ be a finite $m$-sized family of LINEAR functions such that:...
- 11
0
votes
1
answer
130
views
Cascading minimization problems
Hi all. Suppose I have a linear programming problem on the vector variable $x$ that has many solutions and let $U$ be the set of these solutions. Suppose I have a second LP problem on $y \in U$. ...
- 57
3
votes
2
answers
10k
views
linear programming with OR restrictions
Hi all. I have a linear program with the restriction that every variable can be zero or greater than or equal to a positive constant. That is:
minimize: $w^Tx$
subject to: $Ax=b$, $Cx \le d$ and for ...
- 57
0
votes
2
answers
891
views
Find both maximum and minimum values in linear programming problem
Hi all. I have a linear programming problem where I need to find both maximum and minimum values of the objective function. The optimal points are not relevant.
Is there an efficient way to do so?
- 57
19
votes
3
answers
2k
views
A generalization of the triangle counting problem for simple weighted graphs
One nice identity is that $$\operatorname{tr}(A^3)/6$$ counts the number of triangles of a graph with adjacency matrix $A$. It also implies that triangle counting in a graph can be performed in sub-...
- 3,463
1
vote
2
answers
134
views
LP/QP with not-so-constant linear constaints
I have an otherwise standard LP or PSD QP problem as below:
$\min\limits_x {c}' x$ subject to $Ax\leq b$
or
$\min\limits_x \frac{1}{2}{x}' Qx + {c}' x$ subject to $Ax\leq b$
the only exception ...
- 11
0
votes
0
answers
783
views
LP relaxation for ILP\IP (integer linear programming)
I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...
3
votes
1
answer
533
views
Solving a system of linear inequalities
Consider the following system of inequalities:
$Ax=b$;
$x\geq 0$;
A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. How this system can be solved without ...
- 221
18
votes
2
answers
2k
views
What is the largest tensor rank of $n \times n \times n$ tensor?
The tensor rank of a three dimensional array $M[i,j,k], i,j,k\in [1,\ldots,n]$ is the minimal number of vectors $x_i,y_i,z_i$, such that $M=\sum_{i=1}^d x_i\otimes y_i\otimes z_i$.
From dimension ...
- 2,179
3
votes
2
answers
195
views
Determination of rationality and computing a rational parametrization
Suppose I have a hypersurface in $\mathbb{C}P^n$ given by some $f(z_1, \dots, z_{n+1}) = 0.$ Is there an algorithm which returns a rational parametrization if there is one, and "not rational" ...
- 96.4k
5
votes
2
answers
604
views
Sets of vectors related by a rotation
We have a two sets of vectors ($\mathbb{C}^d$), $A=\{ v_1, \ldots v_n\}$ and $B=\{u_1, \ldots u_n\}$.
The question is if there is an efficient solution (polynomial in $n$) for checking whether $A$ ...
- 1,612
0
votes
1
answer
205
views
SDP Algorithms/ maximally complementary solutions
Hello,
I was wondering if there are algorithms for (linear) Semidefinite Programs (SDP) out there, that converge towards a maximally complementary solution, even if strict complementary does not hold.
...
- 1
0
votes
1
answer
504
views
$\ell_o$ Minimization (Minimizing the support of a vector)
I have been looking into the problem
$\min: \|x \|_0$ subject to$: Ax=b$. $\|x \|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time ...
- 11
58
votes
2
answers
18k
views
How fast can we *really* multiply matrices?
Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.807})$ for the multiplication of two $n \times n$ matrices (the exponent is $\frac{\log7}{\log2}$). ...
- 15.5k
2
votes
1
answer
6k
views
sum of maxima vs the maximum of the sum
Consider the following integer program
$$
\begin{align}
\max &\sum\nolimits_{i}\sum\nolimits_{j} U_i(j)\cdot x_{i,j}\\
\text{subject to}& \sum_{i}x_{i,j}\cdot f\left(i,j\right)\leqslant c_j,&...
- 21
0
votes
1
answer
314
views
graph to tree and graph isomorphism problem
Sorry if the following are stupid questions (i do not know much about the graph theory).
1. Motivation
we do not know the graph isomorphism problem in class P or NP complete and it is P in the ...
- 2,773
1
vote
2
answers
242
views
what method can I employ to solve this optimization problem which involves \min?
The optimization problem is:
maximize $$\min(\sum\limits_{i=1}^N \log\left(a_{1,i}+\frac{b_{1,i}}{c_{1,i}+d_{1,i}x_i}\right),\sum\limits_{i=1}^N \log\left(a_{2,i}+\frac{b_{2,i}}{c_{2,i}+d_{2,i}x_i}\...
- 764
4
votes
1
answer
1k
views
Complexity of convex polytope volume calculation ? (Volume of Voronoi cell) (Error probability)
Assume I have polytope in R^k given by N (k<< N) linear inequalities (A_i x < b_i).
I guess complexity of its volume calculate is higher than linear in "N", am I right ?
(Is the complexity ...
- 24.9k
4
votes
2
answers
4k
views
Dual Norm For Sum of 2-Norms
What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$
$\|\mathbf{x}\| = ...
1
vote
2
answers
660
views
constructing a curve dividing two sets of points
Lets assume I have two sets of points, each characterized as being "A" or "B", respectively, that are in a Euclidean plane. Theoretically these two sets are samplings from a space that has ...
- 11
17
votes
1
answer
2k
views
Forcing over set theory versus forcing over arithmetic
I've been trying to understand better some of the research on forcing over bounded arithmetic and its connections with lower bounds in complexity theory. For example, Takeuti and Yasumoto have some ...
- 82.7k
2
votes
3
answers
2k
views
Efficient Algorithm For Projection Onto A Convex Set
Given $\mathbf{x} \in \mathbb{R}^n$ and $\tau$ a scalar, I would like to solve the following Euclidean projection problem:
$\underset{\mathbf{p}}{\mathrm{argmin}} \; \|\mathbf{p}-\mathbf{x}\|_2
\;\;
\...
3
votes
1
answer
2k
views
Oracle Results: P^A = NP^A
Context
In the work of Baker, Gill, Solovay, we know that there exists some oracle A s.t.
$$P^A = NP^A$$.
Now, in CCAMA, this oracle $A$ is given as an EXP complete language.
Question:
Can we do ...
- 45
2
votes
0
answers
230
views
Consistency of a system of linear equations
I have a system of linear equations in form of $AX=b$ where $A_{m\times n}$, $X_{n\times 1}$ and $b_{m\times 1}$. Coefficient matrix $A$ is quite sparse. However, using a practical LP solver like ...
- 221
1
vote
0
answers
89
views
Deciding / Approximating Parity of Small Depth Decision Trees
Let C be a circuit such that:
C: $\{0,1\}^n$ to $\{0,1\}$
the top most gate is a parity gate
all the inputs to the parity gate are small depth decision trees
there is a total of $2^{ log^k n}$ ...
- 45
1
vote
0
answers
628
views
Totally unimodular Matrices
A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...
- 11
7
votes
2
answers
7k
views
Computational complexity of calculating the nth root of a real number
Several sources state that the computational or time complexity of square rooting is the same as that of multiplication (or division). See for example:
Jean-Michel Muller, "Elementary Functions: ...
- 385
1
vote
3
answers
5k
views
How to get the largest subset of a set of sets of intervals with no overlapping intervals
Given: Set of Set of Intervals. Example {{(1,2), (3,4)}, {(1, 3)}, {(13,14)}}
Wanted Result: Largest Subset, in which no Interval overlaps with another from every Set pairwise.
Example:
Input {{(1,...
- 41
3
votes
1
answer
347
views
Grading a non-graded poset as squeezed as possible
Here is a curiosity question (motivated by the recent revamp of ranked-poset routines in Sage).
Let $P$ be a finite poset. We look for a family $\left(a_p\right)_{p\in P}$ of real numbers summing up ...
- 33.8k
2
votes
3
answers
398
views
Generating a set of integer passwords that can be securely authenticated
First, apologies for the title. This is an odd question, and I couldn't come up with a simple title for it.
My question is as follows.
Given a positive integer $k$, determine a set of properties $S$ ...
- 135
6
votes
2
answers
1k
views
Computational complexity of Knot polynomials
What's known about computational complexity of different types of knot invariant polynomials?
For example, Evaluating Jones Polynomial is known to be #P hard.
Is there any reference that surveys such ...
- 615
6
votes
3
answers
1k
views
computational complexity of primitive recursive functions
If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That ...
- 63
10
votes
1
answer
411
views
Network flows with capacities on pairs of edges
Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$.
Now add edge-pair ...
- 37.7k
6
votes
2
answers
518
views
A minimum set hitting every base of a matroid
We are given a matroid. Our goal is to find a set of elements of minimum size that has non-empty intersection with every base of the matroid. Is the problem studied before? Is it in P? For example, in ...
- 115
0
votes
1
answer
226
views
Continuity of Lexicographic Minimum Solution of a parametrized LP problem
Given a parametrized LP problem
find x, that minimizes F*x
such that Ax <=Bt+D
where t is a parameter.
And suppose C(t) is a set of all optimal solutions of LP with parameter t.
Let x_L(t) be ...
- 29
4
votes
2
answers
280
views
How small can a language in NP\P be?
How small can a language in $NP$ but not in $P$ be? Of course, I don't expect a proof that there exists a language in $NP\setminus P$, so instead I'll ask: Can we rule out any of these conjectures?
1)...
- 1,593