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3
votes
0answers
82 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 ...
0
votes
0answers
18 views

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 ...
2
votes
1answer
102 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 ...
2
votes
1answer
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, ...
1
vote
0answers
64 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?
2
votes
0answers
111 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}. ...
13
votes
1answer
314 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 ...
4
votes
2answers
186 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} ...
1
vote
1answer
52 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,\ \ ...
2
votes
0answers
197 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 ...
1
vote
0answers
123 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 ...
1
vote
2answers
142 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. ...
3
votes
1answer
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 $x_i$. ...
4
votes
0answers
80 views

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 ...
4
votes
3answers
218 views

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
0
votes
2answers
290 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 ...
1
vote
1answer
95 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 ...
0
votes
0answers
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 ...
0
votes
0answers
94 views

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 ...
3
votes
1answer
395 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 ...
1
vote
0answers
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 ...
1
vote
0answers
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? ...
0
votes
2answers
242 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 ...
2
votes
1answer
134 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 ...
6
votes
1answer
346 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 ...
1
vote
1answer
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? ...
2
votes
2answers
166 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. ...
2
votes
1answer
174 views

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 ...
7
votes
2answers
340 views

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 ...
-1
votes
1answer
54 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 ...
2
votes
0answers
63 views

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 $\bf{x}$ ...
1
vote
1answer
115 views

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< ...
3
votes
1answer
213 views

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 ...
1
vote
1answer
80 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 $k$ ...
6
votes
1answer
708 views

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 ...
2
votes
1answer
442 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 ...
0
votes
0answers
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} ...
0
votes
0answers
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 ...
4
votes
3answers
568 views

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 ...
1
vote
2answers
269 views

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: ...
1
vote
0answers
84 views

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 $$ ...
10
votes
1answer
667 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 ...
0
votes
1answer
593 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 ...
3
votes
0answers
390 views

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 ...
2
votes
1answer
138 views

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 ...
6
votes
1answer
389 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 ...
4
votes
0answers
131 views

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 ...
6
votes
1answer
412 views

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 ...
14
votes
0answers
601 views

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 ...
2
votes
2answers
305 views

High dimensional Steiner tree

Given n affinely independent points in n-1 dimensional Euclidean space, how is the minimum Steiner tree constructed? Or assuming that the topology of the Steiner tree is given, is there an easy way ...