The combinatorial-optimizatio tag has no wiki summary.

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### Given $M$, minimize $|Mx|_0$

Given a matrix $M \in \mathbb{R}^{n \times m}$, I would like to find
$\min_{x \in \mathbb{R}^m} \|Mx\|_0$ such that $x \neq 0^m$,
where the $\ell_0$ "norm" is measured by simply counting the number ...

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100 views

### What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...

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### Complexity of optimizing a bi-objective function with integer constraints

I have two different objective functions, each of which can be solved optimally in polynomial time. Does this mean, I can optimize a linear combination of these objective functions in polynomial time ...

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106 views

### Optimization over symmetric polynomials

Consider the following constraint satisfaction problem: Let $\alpha_1 , \ldots, \alpha_k \in \mathbb{R}$ be given as well as an error parameter $\epsilon$. Find $p_1, \ldots, p_n$ such that
(i) $0 ...

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**1**answer

107 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, ...

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66 views

### Limit shape for oil-shaped stack in the max overhang problem

In the Maximum Overhang paper, the authors mention an oil-shaped configuration (ref. page 19)
What is known about the curve that limits this shape?

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118 views

### Solution of a linearly constrained quadratic programming problem [closed]

What is the solution of the following optimization problem:
\begin{align}
&\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\
&\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}.
...

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324 views

### How to roll a $p$

Let $p$ be a positive integer (which is not a power of $2$), and suppose we want to generate a number uniformly randomly in the set $\{ 0, 1, \dots , p-1 \}$ (to emulate a dice roll). We are given ...

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197 views

### Complexity of finding the maximum sum divided by product

What is the complexity of the following optimization problem?
Problem.
Given $n$ pairs of positive reals $(a_i,b_i)_{i=1}^n$, choose a subset $S \subseteq [n]$ to maximize
$$
\frac{\sum_{i\in S} ...

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**1**answer

56 views

### Maximizing a certain concave function over a non-convex set

I have been working on a problem which involves maximizing a concave function over a non convex constraint set. The problem is the following $$max. \frac{1}{2}x^TBx\\ s.t.\ x_i(1-x_i)=0,\ \ ...

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198 views

### Minimizing $\{0,1\}$-sequence permutations

Explanation: For a given bit sequence $f$, reposition the bits as to minimize $G$ which can be thought of as a measure of how poorly proportional $f$ is to each of its subsequences.
Let $p \in ...

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125 views

### current status of combinatorial optimization solvers [closed]

What is the current status of the solvers in combinatorial optimization? For example, what is the "usual feasible size" for a traveller sale's man problem, say, how many nodes and edges are "usually ...

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145 views

### Combinatorial optimization problem involving infinite spin system

In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity.
...

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108 views

### A difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$.
(Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...

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### Nice minimal embeddings of large finite groups into compact Riemannian manifolds

The initial motivation for this question is a very practical problem: I need to find the absolute minimum of a function on a very large symmetric group $\Sigma_N$ (with $N$ 10000 or more). So if ...

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### Can anyone suggest a text on polyhedral theory?

Can anyone suggest a text on polyhedral theory? Particularly on increasing the number of faces under projections. 0,1 polytopes

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**2**answers

293 views

### Mixed integer programming formulation for Ising model

I want to implement a minimisation on a 2D spin Ising model with 30x30 grid. The spin variables is 0,1 and the objective is to minimize the sum of products of spins. For simplicity, I only include NN ...

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**1**answer

96 views

### A certain instance of the Set Covering problem

Is there any useful structure associated with the following instance of the Set Covering problem?
Let $G$ be a weighted graph and let $\mathcal{P}$ denote the set of all shortest paths between all ...

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58 views

### Sufficient optimality condition for a non-smooth quasiconvex problem

The result of relaxing to an integer program is the following optimization problem:
$$\min_{\textbf{x}} \sum_{i=1}^n \alpha_i h(x_i)\quad subject \; to \quad A\textbf{x} = \textbf{0}$$
where ...

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### Ref request: If an affine subspace $V$ of $\mathbb R^n$ meets an $n$-dimensional polytope $P$, then it meets $P$ in a face of dimension $\le n-\dim V$

Let $P$ and $V$ be, respectively, a bounded full-dimensional polytope and an $m$-dimensional affine subspace of $\mathbb R^n$, and assume $P \cap V$ is non-empty.
Q1. Could you provide me with a ...

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401 views

### Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality

The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...

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64 views

### integrality of a linear program — binary equality constaints

Consider the following linear program:
$\left\{
\begin{array}{l}
\underset{x}{max} \;\;c^Tx\\
[I, \;B]x = \mathbf{1}\\
x\geq 0
\end{array}
\right.$
where $c$ is a vector ...

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62 views

### Are there any known bounds on the value of solutions of linear integer programming?

Given a linear objective function and a system of linear constraints; are there any known bounds on the values of (positive) integral solutions in terms of the coefficient matrix of the constraints?
...

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249 views

### Supermodular Minimization

I need to minimize a supermodular function and I am well aware of the fact that minimizing supermodular functions is equivalent to maximize submodular functions and that there are many good ...

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**1**answer

135 views

### Minimize the length of intersection of the set of intervals

Consider the following problem: given a finite set $S$ of intervals on the line and a number $k$. We need to colour this set in $k$ colours so that the measure of the set of points, which are ...

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357 views

### Best ranking in tournament: polynomial time algorithm?

This question was posed by my colleague Torbjörn Lundh in his paper Which Ball is the Roundest? A Suggested Tournament Stability Index, Journal of Quantitative Analysis in Sports 2(3), 2006. We have ...

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205 views

### Is this Graph parameter known?

Let $\lambda(G)$ denote the edge-connectivity of $G$.
Consider the following parameter:
$\rho(G) = \max_{X \subset V(G)} \min(\lambda(G[X]), \lambda(G[V(G) - X]))$
Has this parameter been studied? ...

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### 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. ...

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### Upper bounds on the worst-case traveling salesman tours in the unit square

The paper [1] proves that, if we place $N$ points in the unit square, then the length $\ell$ of the euclidean TSP tour of those points must satisfy $$\ell \leq \sqrt{2N} + 7/4~~.$$ I'm wondering, can ...

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### Quickly finding optimal subset of pairs of numerator and denominator terms for special objective functions

Given a multi-set of pairs $((a_i,b_i))_{i \in Y}$ of positive numerator and denominator terms (i.e. each pair has one numerator term and one denominator term), my general problem is to find the ...

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55 views

### Effect of a Specific Restriction on the Integrality of Min-cost Flow Solutions

I want to model the following situation: there is one production site (modelled by the source), a collection of depots (modelled by nodes without demand) and, of course many more customers (modelled ...

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### Optimization over a variable domain defined as a convex hull of given points [closed]

I have an optimization problem:
$\max_{\bf{x}} Z(\bf{x})$,
s.t. $\bf{x} \in conv(\bf{S})$
where $\bf{x}$ is an $n$-dimensional vector, $Z(\bf{x})$ is a non-linear function. The domain of $\bf{x}$ ...

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### Deducing Linear Inequalities

Let $X_1,X_2,\ldots,X_n $ be indeterminates. Denote by $S$ the set of all linear inequalities of the form
$X_{i_1}+X_{i_2}+\ldots+X_{i_k} \geq k,$
with $k \in \{ 1,2,\ldots,n \}$ and $1 \leq i_1< ...

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### Optimization problem - maximizing number of satisfied linear inequalities subject to a quadratic constraint

I am wondering what is known about optimization problems of the following type.
Our control x is a unit vector in $\mathbb{R}^n$. We are given a finite number of linear inequalities
$$Az≥b,$$
and we ...

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**1**answer

84 views

### Finding a sub-matrix from a fat matrix with the best condition number.

Given a m-by-n matrix with $n>>m$ and with a known rank of $k\leq m$, what would be a computationally effective way of finding out $k$ columns, such that the matrix formed using these $k$ ...

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### Inverse of a totally unimodular matrix

A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...

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461 views

### Maximizing supermodular functions

I have a real supermodular objective function which I want to maximize with constraint. The constraint is on the size, like |A|=k .
I am wondering if anyone can give me more information about a ...

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60 views

### Approximation for accumulative set cover

Let $S_1,\ldots,S_m\subseteq U$ be subsets of a set $U$ of size $\lvert U\rvert=n$. Over all permutations $\pi$ of the set $\{1,\ldots,m\}$, I want to maximize the quantity
\begin{equation}
...

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69 views

### Laplacian using SDP

Is there any suggestion about how could one construct a model that uses semidefinite programming that minimizes sum of k smallest eigenvalues of Laplacian matrix?
I found two papers that have done ...

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### Selecting $k$ integers from an interval $[0, N]$ to maximize the minimum difference between pairwise sums

I have an optimization problem where I need to select $k$ integers over the interval $[0, N]$ s.t. I maximize the minimum difference between any pairwise sum of the $k$ integers (where we also include ...

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### sorting two paired lists of real numbers to minimize consecutive absolute differences

Consider a set of $n$ real-valued number pairs: $(x_1,y_1), (x_2,y_2), \dots, (x_n,y_n)$. I want to find a permutation $p$ of the indices which minimizes the sum of consecutive absolute differences: ...

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### Maximum magnitude subset sum

Let $z_1,z_2,\dots,z_N$ be vectors from $\mathbb Z^m$ for some $m$. The problem is:
Given a positive integer $p$, find the subset $A_p \subset \{ 1,2,\dots,N \}$ of size $|A_p| = p$ such that
$$
...

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672 views

### Lotteries, Turan's problem, and minimization of risk

Suppose I am a high-volume broker aiming to make some money on a state lottery. In this lottery, six balls are drawn from a population of (let's say) 50, without replacement. A ticket is a choice of ...

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620 views

### Finding the lowest cost set of disjoint paths using all nodes in a directed graph?

I have a directed graph with edges connecting nodes representing costs.
I wish to find the set of paths which
-go from node 'start' to node 'end'
-are node-disjoint (except for the start and end ...

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### Minimum weight bipartite graph clique covering

I was wondering if anyone here could give me any pointers as to how to solve the following problem.
Let $B=(L,R,E)$ be a bipartite graph, and $\forall u\in L\cup R$, let $c_u$ be a cost associated to ...

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### set and subset series combinatorics

let $A=\{1,2,3...,N\}$ and $B_1,B_2,B_3\dots,B_n$ be a series of subsets of $A$ which satisfied that $|B_i|=m$,
$|B_i\cap B_j|\le k$. what is the maximum of $n$? ($k< m< N$)
it can be easily ...

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393 views

### How does this algorithmic proof of Edmonds-Gallai work?

Sorry, this is going to be technical and dirty. I am not looking for a proof of the Edmonds-Gallai structure theorem (I understand two of them, even if they are rather similar); I am trying to ...

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### Convexified threshold of a function

Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set.
It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when ...

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### Wrapping a convex polyhedron with string

This is a meta-question, rather than a specific mathematical question.
I am seeking a mathematical definition that captures the following physical idea.
Suppose you have a convex polyhedron $P ...

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### A Conjecture About Directed Graphs that are the Union of Two Trees

Let D=(V,E) be a directed graph that is the union of two edge-disjoint directed
spanning trees. Suppose that
There no subset X of vertices so that
there is precisely one directed edge
from X ...