All Questions
598 questions with no upvoted or accepted answers
1
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89
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Polynomial time for a quadratic equation and linear inequalities?
Does anyone know how to find a feasible solution (or the infeasibility of any solution) in a polynomial time to the following problem:
\begin{align*}
xAx^t = 0, \\
Bx^t = c, \\
x_i \ge 0,
\end{align*}
...
- 11
1
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0
answers
68
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Fundamental regions in convex programming
In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
- 13.9k
1
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0
answers
147
views
Convergence of infinite linear programming
Suppose we have the following linear program (LP1),
$$\min_{f \in \mathcal{C}} \int_{\mathbb{R}} f(t) \cos(2 \pi x_0 t) dt \\ \text{subject to } \int_{\mathbb{R}}f(t)dt = 1 \\\forall x\in [0,1]: \int_{...
- 53
1
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0
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104
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Meaning of L-reduction from Dominating set problem
We are working in a variation of Locating dominating sets. Recently, we realized that the reduction from dominating set to our problem in proving its NP-completeness turns out to be also an L-...
- 11
1
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0
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82
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What is the relation between different generalizations of linear programming?
Linear programming subsumed by each of
Semidefinite programming (SDP)
Convex programming (CXP)
SOS programming (SSP)
Is there any relation between each pair in the three?
Are all three equivalent in ...
- 1,826
1
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0
answers
144
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Checking existence of proofs of fixed length
This question is a continuation of a related previous question (check here).
Let $\mathcal{L}$ be a recursive first-order theory with the Hilbert-Ackerman's proof calculus, and such that the ...
- 3,318
1
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0
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147
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The chromatic polynomial of a line graph
Is there a way to obtain the chromatic polynomial of the line graph of a regular simple graph, having known the chromatic polynomial of the graph?
There already exist characterizations of line graph ...
- 2,089
1
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0
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78
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Reference for the algorithm to find the intersection between a subspace and positive orthant
I came across this algorithm, in this question Algorithm for the intersection of a vector subspace with a cone of non-negative vectors ;
Is there any reference for the algorithm described in the ...
1
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0
answers
67
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What are the corners of this polytope?
Let $f$ be a non-negative function on the positive integers such that $f(s+t)\geq f(s) + f(t)$ for all $s,t\in\mathbb{Z}^+$. Consider the polytope consisting of all $x\in \mathbb{R}^n$ such that $$\...
- 11
1
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0
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28
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Modified straightline complexity of almost square of sums
Assume every linear operation (such as inner product with constant vector) can be performed in one step and multiplication by variables (quadratic operation) can be performed in one step.
We know the ...
- 1,826
1
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0
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1k
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Computational complexity of computing the trace of a matrix product under some structure
I have two problems related to computing some trace, and some (possibly suboptimal) answers. My question is about a potential more efficient algorithm for each one. (More interested in an answer to ...
- 123
1
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0
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127
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Does relationship between c.e.set, productive set, immune set, ML-random set exist between sets of class of other level
Is relationship between c.e.set, productive set, immune set, ML-random set similar to relationship between polynomial complexity set, polynomial complexity-productive set, P-immune set, P-random set?
- 3,094
1
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0
answers
283
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total unimodularity of a matrix
Let G be the node-arc incidence matrix of a given directed network (rows of $G$ correspond to nodes and its columns correspond to arcs). Let $B_1,\dots, B_K$ denote a partition of the nodes of the ...
- 393
1
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0
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78
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Deciding if a set of hyperplanes passes through a point
There is a set of vectors $\vec a_{k,i}$ with $k\in\{1,\dots,n\}$ and $i\in\{0,1\}$,
such that every $n+1$-tuple $\{\vec v_{1,s_1},\dots,\vec v_{n,s_n},\vec v_{r,1-s_r}\}$
is linearly independent for ...
- 1,207
1
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0
answers
76
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Classically compute ahead time if Lie Algebra is either polynomial finite or geometrically closed?
Given a set of $N$ operators $\mathcal{O}$ with a known set of Lie Algebra group multiplication rules $\mathcal{G}$ that can be programmed into a classical computer, is there a classical poly($N$) ...
- 215
1
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0
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163
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Can we reduce the maximization of this integral to the maximization of the integrand?
I would like to know whether we are able to reduce the following optimization problem to the pointwise optimization of the integrand (or how we can solve it otherwise): Maximize $$\sum_{i\in I}\sum_{j\...
- 167
1
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0
answers
25
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Weird subspace/equality-constrained LP problem/variant of change-making problem
Assume that we have a set, $\mathscr{R}$ containing $m$-dimensional vectors. Solve
$$\sum_{i=1}^n c_i\leq\delta$$
$$\text{subject to } \sum_{i=1}^n r_i c_i=x^\prime \text{ for all }x^\prime$$
where
$0\...
- 11
1
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0
answers
222
views
Way to express a number in its most compact sum of powers
Given a non-negative number n, what is the best approach to find the most compact representation for n in terms of sums of powers, such that the bases and the exponents can't surpass a given value (...
- 11
1
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0
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121
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Calculate amount of FLOPs for an eigenvalue problem solver
I've got 2 complex, non symmetric, matrices $A_{1000x1000}$, $B_{1000x1000}$ and I am using Matlab to get it's eigenvalues (functions like eig or eigs). Both matrices are different - one is more dense ...
- 111
1
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0
answers
68
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Projection of a polytope along 4 orthogonal axes
Consider the following problem:
Given an $\mathcal{H}$-polytope $P$ in $\mathbb{R}^d$ and $4$ orthogonal vectors $v_1, ..., v_4 \in \mathbb{R}^d$, compute the projection of $P$ to the subspace ...
- 11
1
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0
answers
36
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Linear programming with a convergent coefficient
The following linear programming problem
$x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$
has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...
- 19
1
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0
answers
126
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Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
- 1,826
1
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0
answers
57
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On divisibility conditions implying local coprimality conditions
This question is inspired by Bernardo Recaman's question Strings of consecutive integers divisible by 1, 2, 3, ..., N on intervals of $n$ integers being divisible by the integers $1$ through $n$. The ...
- 13k
1
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0
answers
88
views
What is the probability of 'yes' to this likely $coNP$ problem?
Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$.
Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...
- 1,826
1
vote
0
answers
24
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Simple monotonicity property for coordinate descent and linear objective functions
Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for ...
- 4,049
1
vote
0
answers
37
views
Fast certficate of negativity for objective value of mixed-integer linear program
Let $c \in \mathbb R^n$, $A \in \mathbb R^{m \times n}$, $b \in \mathbb R^m$, and $I \subseteq \{1,2,\ldots,n\}$. Consider the Mixed integer linear program (MILP)
$$
\begin{split}
f^* = &\max \; ...
- 6,853
1
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0
answers
67
views
Computational complexity of fractions multiplication puzzle
I developed a puzzle for which I am seeking an EFFICIENT algorithm (I searched Google but did not find this puzzle in the literature, ):
You have $k$ rationals, $n_1/d_1, n_2/d_2, ..., n_k/d_k$.
...
- 2,993
1
vote
0
answers
36
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Is there an algorithm for this constrained Hypergraph optimization problem?
I'm currently developing an algorithm for computing knot coloring invariants and got to the following question:
Given a set $S$ and a certain hyper-graph $H \subseteq S^3 $, find a decomposition $S = ...
- 133
1
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0
answers
29
views
Existence of Costas array with specified displacment vectors?
Costas array is a set of $n$ points lying on the square of a $n×n$ checkerboard, such that each row or column contains only one point, and that all of the $n(n − 1)/2$ displacement vectors between ...
- 2,993
1
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0
answers
70
views
Shortest Lattice Vector with restricted $x$
Let $\Lambda$ be a lattice with basis, $B$ consisting of vectors $b_i$, so that the elements of $\Lambda$ are of form, $y\in \Lambda \iff y=Bx=\sum_i b_ix_i$ for some $x_i\in\mathbb{Z}$.
My questions ...
- 1,096
1
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0
answers
207
views
Is it known whether $\mathrm{NP \subseteq P/poly}$?
It is not immediately clear to me whether this statement is true or false. Can finite restrictions of NP problems be computed in polynomial time?
- 11
1
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0
answers
74
views
How to minimize n-polytope's bounding box with linear transformation?
I am working on an exact algorithm for integer linear programming for my master's thesis:
$Ax\leq b, x \in \mathbb{Z}^n$
$cx\rightarrow min$
For my idea to work out, I need a guarantee that n-...
- 11
1
vote
0
answers
78
views
Bipartite clustering is NP-hard?
Let $G = (A\cup B, E)$ be a bipartite graph with edge weights $w: E\to \mathbb{R}$. Find a partition $B_1, B_2$ of $B$ and a nonempty disjoint subsets $A_1, A_2$ of $A$ such that $w(A_1,B_1) + w(A_2, ...
- 221
1
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0
answers
140
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Is the partition of bipartite graphs NP-hard?
I wonder if the following problem is NP-hard. Is it?
Given a bipartite graph $G = (U, V, E)$ with weights $w : E \to \mathbb{R}_+$, find a partition of $U$ into $U_1, U_2$ and nonempty disjoint ...
- 221
1
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0
answers
66
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On number of solutions by simplex and number of solutions in total in a linear optimization problem?
This is more of a clarification query.
Mizuno http://www2.ims.nus.edu.sg/Programs/012opti/files/talk_mizuno1.pdf says if we give a linear optimization problem
$$\max c'x$$
$$Ax\leq b$$
where $A\in\...
- 13.9k
1
vote
0
answers
62
views
LP Constraints for Bridgeless Cactus Graphs
When trying to determine the optimal bridgeless spanning cactus graph of a weighted, symmetric graph, I got stuck.
What I do not know how to capture, is
the variable number and sizes of the cycles
...
- 13.2k
1
vote
0
answers
60
views
On the defect of a flow network
This problem in graph theory was actually motivated by some problems in Theory of Fractals.
To formulate the problem I need to recall some definitions related to flow network.
A flow network is a ...
- 41.8k
1
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0
answers
86
views
Infinite system of equations with finitely many constraints
During my research I have stumbled upon the following issue concerning infinite systems of linear equations. I do not have much practice in such settings, so I am asking you whether the following ...
- 1,151
1
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0
answers
44
views
In convex optimization we know that the optimum solution is on which hyper plane
We have a standard linear program, I mean a set of inequalities $c_i^Tx\leq b_i$ where $i\in \{1,\ldots ,k\}$ and we want to find $max\{c^Ty| y\in \{\cap \{x|c_i^Tx\leq b_i\}\}$. I put some condition ...
- 19
1
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0
answers
384
views
Complexity of conic optimization problems
I am interested in bounding the computational complexity of the interior points method for solving a generic conic problem of the form
\begin{equation}
\min_x \left\{ c^T x : \mathcal{A}x-B\in\mathbf{...
- 345
1
vote
0
answers
87
views
computing number of primes with congruence constraint
Suppose I give you $(a, b, M)$ and ask you to compute the number of primes not exceeding $M$ which are congruent to $b$ mod $a.$ What is the most efficient way of doing it? (without congruence ...
- 96.4k
1
vote
0
answers
105
views
How is the minimax oracle used to find the oracle complexity of projected subgradient?
I am going through a set of blog posts on the complexity of projected gradient method.
https://blogs.princeton.edu/imabandit/2013/03/15/orf523-oracle-complexity-large-scale-optimization/ defines the ...
- 277
1
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0
answers
261
views
Prove that the following set of triples forms a convex polytope
Take $a,\,b,\,c,\,d \in \mathbb R_+$ such that $a+b+c+d=1$. Define:
\begin{equation}
x_1 = \min(a+b,\,c+d)\,,\qquad x_2 = \min(a+c,\,b+d)\,,\qquad x_3 = \min(a+d,\,b+c)\;.
\end{equation}
I would like ...
- 31
1
vote
0
answers
49
views
Cost associated set problem NP-hard
I have the following problem. I wonder whether or not it appears in the literature. Is it NP-hard?
Given a set $S = \{1,2,\ldots,m\}$, and $A_1,\ldots, A_n$ are subsets of $S$. Each set $A_i$ has ...
- 221
1
vote
0
answers
42
views
Computation of sub-gradient for a concave envelope
Let $x_1<\cdots<x_n$ be $n$ points on real line and $g=(g_1,\cdots, g_n)\in\mathbb R^n$ be the scattered data. Let $u_g: [x_1,x_n]\to\mathbb R$ be the linear interpolation of $g_1,\cdots, g_n$, ...
- 345
1
vote
0
answers
81
views
Maximizing sum of homogeneous functions of order one over a polytope
Let $f_i: \mathbb{R}^n\rightarrow \mathbb{R}$ be
concave, increasing (i.e., if $x\geq y$ where the inequality is entry wise, we have $f_i(x)\geq f_i(y)$), and a
homogeneous function of order one for ...
- 393
1
vote
0
answers
88
views
On convex quadratic programming clarification
We know convex quadratic programming is in $P$.
Is it also in $P$ if the function of interest is only convex in the domain of interest?
- 13.9k
1
vote
0
answers
20
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Calculating Cost-Optimal 1-Factors in Digraphs
I need to find a cost-optimal 1-factor in a positively weighted, directed, regular graph $G(V,A)$ without antiparallel arcs, i.e. given $$\text{deg}_{\text{in}}(u)=\text{deg}_{\text{in}}(v)=\text{deg}...
- 13.2k
1
vote
0
answers
93
views
quick hull algorithm detail
When using quick hull algorithm to find the polytope for half space intersection, we are required to provide an interior point to the solver qhalf.
In other words, providing
$$Ax \le b$$
is not ...
- 867
1
vote
0
answers
43
views
a question about probabilities on spaces of digraphs
Let $G$ be a directed graph with fixed nodes $s$ and $t$. Assume that each edge $e$ in the graph comes with a number $n(e)\in[0,1]$.
We consider probability spaces $S$ whose points are directed ...
- 61