Questions tagged [combinatorial-optimization]
Combinatorial optimization typically deals with optimizing over a finite set of objects that have some combinatorial structure (e.g. trees, matchings, matroids). Approximation algorithms, polyhedral methods, and integer programming are all on topic.
35 questions
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LP Constraints for Connected Subgraphs of Fixed Size
Question:
how can the connectedness-constraint for a subgraph, that is induced by a proper subset $W\subset V$ of the vertices of $G(V,E),\ |V|=n,\ |W|=m$, be formulated in a $LP$ or $ILP$?
...
- 13.2k
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1
answer
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Maximizing the mutual Hamming distance in $\big[{\cal P}([n])\big]^n$
If $X$ is a set and $A,B\subseteq X$ we let the Hamming distance of $A,B$ be defined as $d_H(A,B) = \big|(A\setminus B)\cup(B\setminus A)\big|$. If $\newcommand{\S}{{\cal S}}\S\subseteq {\cal P}(X)$, ...
- 48.9k
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Reliability of ILP approach to number-theoretic optimization
This question is inspired by the recent answer, where @RobPratt proposed to use integer linear programming (ILP) for solving a number-theoretic optimization problem. I will consider a very similar ...
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A question about decomposition of irreducible root system
Fix an irreducible root system $\Phi$ with rank $r$ and a root base $\Delta$ (we only care type ADE). Call a disjoint union $\Phi = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if ...
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3
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What is the smallest size of a shape in which all fixed $n$-polyominos can fit?
Let $n$ be an integer and consider all fixed $n$-polyominos, i.e., without rotation or reflection. I am interested in finding a shape in which all polyominos can embed. (It is OK if multiple ...
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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 to its ...
- 24.7k
11
votes
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A discrete optimization problem related to the AM-GM inequality
Let $k\in\mathbb{Z}_{>0}$, and $s\in\mathbb{N}$, and for $m_1,\ldots,m_k$ some nonnegative integers, consider the problem of maximizing the product
$$
(1+m_1)(1+m_2)\cdots(1+m_k)
$$
under the ...
9
votes
3
answers
446
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Pairs of vertices with high degree difference
Consider a finite, simple and undirected graph $G=(V,E)$ with $V=\{v_1,\dots, v_n\}$. Let us also fix an integer $k> n/2$. What are we able to say about the following quantity:
$$\mathcal{I}_k(G) :=...
user153000
8
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3
answers
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Shrinking subset and product
Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|...
- 1,209
8
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1
answer
403
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Smallest relation in complement of partial order that prohibits its extension
Let $P$ be a partial order on a finite set $S$ (assume that every element is related to at least one other element besides itself…this raises a few quick questions: is this implied by the definition ...
- 870
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1
answer
804
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Combinatorial optimization problem for bipartite graphs
Let $G(V_1\cup V_2, E)$ be a simple bipartite graph having $n$ vertices and $m$ edges, such that $|V_1|=|V_2|$ (which implies that $n$ is an even number). Given any node $i \in V_1\cup V_2$, we denote ...
- 1,677
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votes
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Reference request: maximal determinant of matrices with pairwise orthogonal rows and entries in $\{1, 0, -1\}$
We know that "Hadamard maximal determinant problem" concerns the largest determinant of a matrix of oder $n$ with entries in $\{-1,1\}$ or $\{0, 1\}$. For $n=4k$, it is the Hadamard ...
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2
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Maximize the determinant of Boolean combinations of positive definite matrices
I have the following optimization problem.
$$\begin{array}{ll} \text{maximize} & \det \left(\sum^n_{i=1}z_i W_i \right)\\ \text{subject to} & \sum_{i=1}^n z_i = N\\ & z_i
\in \{0,1\}\end{...
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3
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views
Optimization algorithm sought
Suppose I have $N$ pairs of positive numbers $(a_1, b_1), (a_2, b_2), \dotsc, (a_N, b_N).$ and I want to find a subset of $M$ of them maximizing
$$
\frac{\sum_{j=1}^M a_{i_j}}{\sum_{j=1}^M b_{i_j}}.
$$...
- 96.4k
6
votes
1
answer
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Desargues ten point configuration $D_{10}$ in LaTeX
I want to draw the Desargues configuration $10_3$ in LaTeX using the standard picture environment, which allows only lines with the slopes $n:m$ where $\max\{|n|,|m|\}\le 6$. Is it possible? If not, ...
- 41.8k
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1
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What is the best way to partition the $4$-subsets of $\{1,2,3,\dots,n\}$?
Also asked on MSE: What is the best way to partition the $4$-subsets of $\{1,2,3,\dots,n\}$?.
Consider the set $X = \{1,2,3,\dots,n\}$. Define the collection of all $4$-subsets of $X$ by $$\mathcal A=...
5
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1
answer
332
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On a certain norm of the identity operator on $\mathbb R^2$
$\newcommand\R{\mathbb R}\newcommand\Q{\mathcal Q}$For mutually orthogonal vectors unit vectors $a=[a_1,\dots,a_n]^T$ and $b=[b_1,\dots,b_n]^T$ in $\R^n=\R^{n\times1}$ (so that $n\ge2$) and for all $x=...
- 128k
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2
answers
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Determining the Largest Face of a Simplex
This question is in the vein of my former question Fast Comparing of the Volume of Simplices Defined by Sidelengths, but it has a different twist, that may allow for an easier answer:
Questions:
...
- 13.2k
4
votes
1
answer
363
views
Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box
Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.
Let $\mathcal{S}_C$ be the set ...
- 1,677
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votes
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229
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Optimal colorings
If $G=(V,E)$ is a graph, we call the smallest cardinal $\kappa$ such that there is a coloring map $c:V\to \kappa$ as the chromatic number of $G$ and denote it by $\chi(G)$.
For any coloring $c:V(G) \...
- 48.9k
3
votes
1
answer
395
views
Symmetric distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $x, y, z$ the outcomes of three i.i.d. random variables $X, Y, Z$ with distribution $\mathcal{D}$, sorted in increasing order, ...
- 1,677
3
votes
1
answer
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What is known about this TSP variant?
Euclidian (planar) TSP asks for a tour with the minimum total length. The problem is known to be NP-hard. I am interested in the variant of finding a closed tour with the minimum enclosed area (...
- 2,993
3
votes
0
answers
147
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Understanding why $\frac{\phi^5}{2}$ solves this 3D optimization problem, where $\phi$ is the golden ratio
I would like to understand the deep meaning of a solution which arises from an optimization problem discussed in a paper of mine since it can be simply stated as $\frac{\phi^5}{2}$, where $\phi := \...
- 1,451
3
votes
1
answer
509
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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 ...
- 867
2
votes
1
answer
218
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Probability distribution optimization problem of distances between points in $[0,1]$
Let $\mathcal{D}$ be a probability distribution with support $[0,1]$. Let $X, Y, Z$ three i.i.d. random variables with distribution $\mathcal{D}$, and $T$ a random variable uniformly distributed in $[...
- 1,677
2
votes
1
answer
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Algorithm for finding minimally overlapping paths in a graph
I'm curious to find an algorithm that solves the following graph-theory problem.
Suppose I have a graph $G(V,E)$ with two disjoint sets of vertices, $V_a$ and $V_b$.
My goal is to find paths from ...
- 25
2
votes
0
answers
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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 ...
- 867
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votes
1
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Description of Linear Time Algorithm for TSP in Halin Graphs
I am looking for a description of the linear time algorithm for the TSP in Halin graphs, that was given in
"G. Cornuejols, D. Naddef, and W.R. Pulleyblank. Halin graphs and the travelling ...
- 13.2k
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vote
0
answers
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Optimal covering trails for every $k$-dimensional cubic lattice $\mathbb{N}^k := \{(x_1, x_2, \dots, x_n) : x_i \in \mathbb{N} \wedge n \geq 3\}$
After a dozen years spent investigating this particular class of problems, I finally give up and I wish to ask you if any improvement is achievable from here on.
The general problem is as follows:
Let ...
- 1,451
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vote
2
answers
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How to solve the optimization problem $\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$?
I am looking for an algorithm to solve the following optimization problem
$$\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$$
where $\mathbf{w}$ and each $\mathbf{x}_i\in\mathbb{R}^d$.
...
- 175
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vote
0
answers
170
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How to solve this matrix optimization problem? [closed]
$$\min_{X,P} \| X D + (P X P') \, D_0 \|_{\text{F}}$$
where $X$ is a symmetric matrix and $P$ is a permutation matrix. $D$ and $D_0$ are given symmetric matrices. Is there some way to solve this ...
- 29
1
vote
0
answers
54
views
Reductions to the MAX-3-DCC Problem
I am currently working on the Max-3-DCC problem that asks for the heaviest vertex-disjoint cycle cover of weighted directed graphs.
The problem has been reduced to 3-SAT in 1979 by L. Valiant in his ...
- 13.2k
1
vote
1
answer
536
views
Finding an optimal covering trail for the set $\{0,1,2,3\}\times\{0,1,2,3\}\times\{0,1,2,3\}$
Here is a key question (i.e., Question 2 below) that, if correctly answered, would let me support a very general conjecture on a wide class of related problems, a conjecture that I have never shared ...
- 1,451
1
vote
1
answer
117
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Shrinking subset with disjoint unions
Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be disjoint finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that ...
- 1,209
0
votes
1
answer
221
views
Minimize overlap penalty between paths in graph
Suppose we have a weighted undirected graph $G(V,E)$. We are given the information that $V_a \cap V_b = \emptyset$ and $V_a,V_b \subset V$.
We want to find paths from all vertices in $V_a$ to all ...
- 25