All Questions
1,809 questions
2
votes
1
answer
130
views
Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?
In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...
- 121
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
2
votes
2
answers
402
views
Maximization of a matrix product by iterative methods
This might not be very difficult, but I think I may have gotten a little confused.
Suppose we are given a matrix A, and would like to find the vector x of modulus 1 which maximises the product xt A x ...
- 949
2
votes
1
answer
362
views
Selecting k sub-posets
I ran into the following algorithmic problem while experimenting with classification algorithms. Elements are classified into a polyhierarchy, what I understand to be a poset with a single root ("...
- 123
2
votes
2
answers
249
views
Indexing schemes of binary sequences
I am looking for "low-complexity" indexing methods to enumerate binary sequences of a given length and a given weight.
Formally, let $T_k^n = \{x_1^n \in \{0,1\}^n: \sum_{i=1}^n x_i = k\}$. How to ...
- 1,839
2
votes
2
answers
362
views
Decision problem restricted to inputs that satisfy some necessary condition.
Consider the following decision problem:
Problem 1
INPUT: A graph G.
OUTPUT: YES if G is 3-colorable, NO if not.
This is a well-known NP-complete problem. Now suppose that we have a necessary (but ...
- 726
2
votes
2
answers
436
views
NP Complete for range sum constraints?
Is the following problem NP Complete?
We have $n$ variables $x_1$,$x_2$,....,$x_n$ and a set of constraints:
$\sum_{i=a_1}^{b_1}x_i = h_1$
$\sum_{i=a_2}^{b_2}x_i = h_2$
$\sum_{i=a_3}^{b_3}x_i = ...
- 21
2
votes
1
answer
216
views
Slicing bivariate exponential generating functions on x and y
Let $F(x, y) = e^{y D(x)}$ be a generating function for sets of objects enumerated by $D(x)$ that also keeps track of the number of sets (enumerated by the variable $y$, while $x$ enumerates the total ...
- 1,171
2
votes
1
answer
209
views
Computational complexity and commuting functions, examples and conjectures
History of the question. I was proposing a conjecture here, called Prop. 1. Fedor Pakhomov showed a counter-example. Here I am proposing a slightly weaker version of the conjecture, Prop. 2, that ...
- 469
2
votes
1
answer
227
views
Solving linear programming without solving linear programming
Let $v_1, \cdots, v_n$ be vectors in $\mathbb R^k$, and let $M$ be the Gram matrix of them.
It's possible to determine from $M$ and $k$ whether the only vector that has nonnegative inner product with ...
- 9,501
2
votes
1
answer
645
views
How to maximise infinity norm of $x$ with constraint $Ax \le b$ using linear program? [closed]
I want to maximise the infinity norm of $x$, subject to constraint: $Ax \le b$. I think you can use a linear program to solve this, but how do you go about formulating it?
2
votes
1
answer
278
views
Fast inverse of asymmetric diagonally dominant matrix with diagonal 1 and non-positive off-diagonals
I am interested in ways to obtain (even approximately) the inverse of an asymmetric diagonally dominant matrix with diagonal 1 and non-positive off-diagonals.
Formally, let $A$ be a $n\times n$ matrix ...
- 59
2
votes
1
answer
139
views
linear programming with $n$ choose $r$ variables
Given parameters $r < n$, define $m = {n \choose r}$ and let $A$ be the $n\times m$ matrix whose columns are all the vectors with $r$ $1$'s and $n-r$ $0$'s. Let $b$ be a positive $n$-vector. Is ...
- 51
2
votes
1
answer
200
views
Two from cubic subgraph hardness
The Problem
For a given graph $G$, the cubic subgraph problem asks if there is a subgraph where every vertex has degree 3.
The cubic subgraph problem is NP-hard even in bipartite planar graphs with ...
2
votes
1
answer
270
views
What optimization problems have solutions with few nonzeros?
Consider the following optimization problem, with $n$ variables and $m$ linear constraints:
\begin{align}
\text{maximize} && c_1 x_1 + \cdots + c_n x_n &
\\
\text{subject to} && a_{...
- 3,549
2
votes
1
answer
264
views
Are there any continuous-time stochastic processes in which transition probabilities are discontinuous functions over time?
In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also ...
- 21
2
votes
1
answer
91
views
Linear program with one quadratic condition convex in domain of interest polynomial time solvable?
$c\leq xy$ is not a convex condition.
However we know $c\leq xy$ is convex in domain $x,y>0$ in $\mathbb R^2$ for any fixed $c\in\mathbb R$.
Is $c\leq x_1y_1+\dots+x_ny_n$ with $0\leq x_1,\dots,...
- 13.9k
2
votes
1
answer
173
views
Generating an arbitrarily long sequence with decreasing Kolmogorov complexity of terms
Is there an algorithm which, given a string $s$, generates a sequence of $|s|$ strings, such that it can be proven in some axiomatic system $S$, that the Kolmogorov complexity of each successive ...
- 851
2
votes
2
answers
403
views
Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative
$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.
What should $A$ satisfy to guarantee the equation set have only zero solution?
- 23
2
votes
1
answer
148
views
Fast algorithm for large-scale, asymmetric transportation linear program
I have a large-ish instance of a transportation problem that is very asymmetric, say of dimensions $100\times10000$. I am currently solving it with a stock LP solver, but obviously something like the ...
- 4,049
2
votes
1
answer
509
views
Under what condition does Courant–Fischer–Weyl min-max principle hold in general?
From Wikipedia:
Let $A$ be an $n \times n$ Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient $R_A :
\mathbf C^n \setminus \{0\} \to \...
- 155
2
votes
1
answer
171
views
Maximization of Binary Multilinear Fractional Function
Problem: Let $a_{i,j}$, $b_{i,j}\in\mathbb{R}$ for all $(i,j)\in\left[m\right]^2$ such that $a_{i,j}=a_{j,i}$ and $b_{i,j}=b_{j,i}$. Let $z_k\in\{0,1\}$ for $k\in\left[m\right]$. We wish to maximize,
...
2
votes
1
answer
530
views
Integer programming and Groebner basis
I enjoyed reading different papers about using Groebner basis to solve integer programming.
Is there any literature about the complexity and/or comparison with other (more classical) methods like ...
- 337
2
votes
1
answer
3k
views
max-flow at max-cost
I have a flow network with gains. In practical terms, a gain is the opposite of a cost. So, I interested in finding the maximal gain of a network flow, what could be interpreted as finding a maximum ...
- 121
2
votes
1
answer
306
views
About Renyi entropy
If one is given a joint probability distribution over a finite set of discrete random variables then I guess there a notion of $\alpha-$Renyi entropy defined for it as $S_\alpha (X_1,..,X_n) = \frac{...
- 2,246
2
votes
1
answer
188
views
$0/1$ programming multiple quadratic constraints
If we have an $n$-variable rank $n$-linear system it is clear we can find whether there exists a $0/1$ solution in polynomial time.
If we have an $n$-variable degree $2$ system how many constraints ...
- 13.9k
2
votes
2
answers
624
views
Time Hierarchy Theorem and P vs NP
One obvious strategy for proving P not equal to NP would be to show that there is some problem in NP which is hard for a time class strictly containing P (the origin of this question is the recent ...
- 99
2
votes
1
answer
229
views
Algorithm to find the vertices of the equidistant lines between N closed polygonal lines
I have a set $\{C_1, C_2, \ldots, C_N\}$ of $N$ nonintersecting closed piecewise linear curves on the Euclidean plane. For every point $x \in \mathbb{R}^2$ we say it belongs to a territory serviced by ...
- 329
2
votes
1
answer
426
views
Network flows with shared capacities
Suppose we have a flow network, with capacity constraints on weighted sums of arc flows, such as:
$$2 f(1, 2) + 3 f(4, 5) + f(3, 7) \leq 10,$$
where $f(1, 2)$ denotes the flow through arc $(1, 2)$....
- 23
2
votes
1
answer
104
views
Standard names and methods for this type of fitting minimization
In material science research, we have come across the following type of problem.
Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization
$$\eqalign{
...
- 867
2
votes
1
answer
301
views
books on very large scale linear optimization
Recently in my material science research, I have encountered problems of very large scale linear optimization. I read the introductory book "Introduction to Linear Optimization (Athena Scientific ...
- 867
2
votes
1
answer
208
views
QBF of exponential length?
We consider a slightly extended version of a nondeterministic finite automaton, call it a "propositional nondeterministic finite automaton". It is defined as follows. Consider a fixed propositional ...
- 125
2
votes
1
answer
449
views
Why can't there be a problem both in P and NPC [closed]
In this illustration, P and NPC are two disjoint set.
We know that NPC is non-empty. If P $\cap$ NPC $=\varnothing$, then there are elements in NP which are not in P. Doesn't this imply that P $\neq$...
- 121
2
votes
1
answer
134
views
Integer point in a non-empty polytope
I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...
- 243
2
votes
2
answers
842
views
Finding the maximum of a multivariate polynomial of degree one
I need to find the global maximum of the function
\begin{align}
f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\
&+\ldots \\
&+ p_n ...
user24451
2
votes
1
answer
359
views
Dimension independent computational complexity of singular value decomposition
Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is ...
- 45
2
votes
2
answers
799
views
Survey on Compared Running Time: Ellipsoid Method vs. Simplex Method
If you look through papers on the Ellipsoid Method, there is a large agreement, that the Ellipsoid Method, although theoretically polynomial, is in practice way slower than the Simplex Method. ...
- 329
2
votes
1
answer
132
views
Computation of the mean of a random variable to estimate algorithm complexity
I made an incremental algorithm which I would like to evaluate the complexity. The algorithm works with a sliding window of size n.
To study the complexity, the window is considered full and the data ...
- 23
2
votes
1
answer
1k
views
Finding integer points inside of a parallelogram
Suppose $P = \{p_1,\ldots,p_4\} \in \mathbb{R}^2$ defines a quadrilateral (here, specifically, a parallelogram). In the particular case I'm dealing with, I know that there exists at least one point ...
- 1,318
2
votes
2
answers
129
views
LP constraint enconding
I have an objective function to be maximized
$obj(x) = \sum_i \gamma_i x_i$ with $x_i \in \mathbb{R}$
With multiple constraints of the form:
$\min_{y \in 0,1} (\sum_{i \in A} \alpha_i x_i + \sum_{i ...
- 21
2
votes
1
answer
227
views
Typical dimension of partial derivatives
Let $V$ be the space of all homogenous polynomials over $\mathbb{C}$ in $n$ variables of degree $d$.
Let $l,k$ be two integers and $f\in V$.
Let $\partial^{=k}(f)$ be the space of all partial ...
- 2,179
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
2
votes
1
answer
403
views
Finding a 5-cycle in a sparse graph efficiently.
Hi,
I was reading this thread: Finding a cycle of fixed length
I want to find a 5-cycle in a graph. Actually, what I really want is a shortest odd cycle of length at least 5, but maybe that is a ...
- 1,834
2
votes
1
answer
1k
views
Probability of system failure in a distributed network
I am trying to build a mathematical model of the availability of a file in a distributed file-system. The system works like this: a node $x$ stores a file $f$ (encoed using erasure codes) at $rb$ ...
- 121
2
votes
0
answers
95
views
Why cannot we adapt Barvinok type counting techniques to general convex integer programs?
Decision problems in Integer Linear Programming have Lenstra type algorithms (https://www.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf) have been generalized to convex integer program ...
- 13.9k
2
votes
0
answers
78
views
Is this variant of post correspondence problem undecidable?
The post correspondence problem, as defined by wikipedia, is undecidable. The problem is defined as follows.
Let $A$ be an alphabet with at least two symbols. The input of the problem consists of ...
- 21
2
votes
0
answers
93
views
How are moduli spaces related to geometric complexity theory?
I am interested in understanding the relationship between moduli spaces and geometric complexity theory (GCT).
Relation between moduli spaces and GCT:
How are moduli spaces related to geometric ...
- 29
2
votes
0
answers
71
views
Lexicographically largest incidence matrix
I have simple algorithmic question, but I can't find any source where this algorithm is explained in details.
Let's assume that we have incidence (with 0 and 1 values) matrix of size $m\times n$. Let ...
- 501
2
votes
0
answers
116
views
Reference for a coarse complexity notion
Throughout, I'm only interested in structures with domain $\mathbb{N}$, no primitive relations, and at least $0,\mathsf{Succ}$ as primitive functions. The length of $m\in\mathbb{N}$ is $\lfloor 1+\...
- 20.5k
2
votes
0
answers
173
views
NP-hardness of a string transformation problem with k templates
Given strings $x$ and $y$, a template length $l$, and a maximum number of different templates $k$, the task is to determine if it's possible to convert $x$ into $y$ using no more than $k$ different ...
- 21