# 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.

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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, ...
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### Real-world datasets for testing the maximum edge bi-clique problem

We have proposed a new approach to solve the maximum edge bi-clique problem, however, we couldn't succeed to find real-world datasets (graph or bipartite graph datasets) to test our approach. Does ...
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### Algorithm for finding a minimum weight circuit in a weighted binary matroid

For a given weighted graph $G = (V, E)$, there is a simple algorithm for finding the minimum weight circuit by running Dijkstra's algorithm $|E|$ times. Also for a matroid $M = (E, I)$ one can use the ...
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### Conjecture on the unsolvability of the $\{3 \times 3 \times \cdots \times 3\} \subseteq \mathbb{R}^k$ dots problem starting from the central point

In 2020 (see Solving the $106$ years old $3^k$ points problem with the clockwise-algorithm, JFMA, 3(2), p. 96), I conjectured that, in the Euclidean space $\mathbb{R}^k$, we can cover any given set of ...
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### Probabilistic optimization problem on tree vertex selection without replacement proportional to the degree

We are given a tree $T(V,E)$ with $|V|=n$ vertices, where $V=\{v_1,v_2,\ldots, v_n\}$. We denote by $d_i$ the degree of vertex $v_i$ for all $i\in\{1,2,\ldots,n\}$. In a sequential fashion, we select ...
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### Boolean operation on n dimensional polyhedron

A polyhedron in $R^n$ is defined by a set of half-planes: $P = \{x \in R^n \mid Ax - b \le 0\}$. Given a set of polyhedra in $R^n$, $P_1, P_2, \dotsc, P_k$, is there an algorithm/implementation that ...
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### Optimization problem on randomly selecting subintervals from a given interval with combinatorial constraints

We select uniformly at random $k$ pairwise disjoint intervals from a given interval $[0,s]$ with length respectively equal to $\ell_1, \ell_2, \ldots, \ell_k\$, i.e., we select uniformly at random $k$...
1 vote
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### 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 ...
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### Dual equivalence of minimum feedback-vertex sets and cycle packing

it is known that the duals of feedback-set problems are set-packing problems; in the context of digraphs the feedback set are either a minimal set of vertices or edges that hit every oriented cycle; ...
1 vote
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### Choosing $k$ different assignments of binary variables in order to capture the largest volume of the joint probability distribution

Assume you have $n$ independent binary variables $\{x_1,\dots,x_n\}$ and for each variable $x_i$ you know that its value is equal to $1$ with a probability $p_i$. I would like to enumerate the joint ...
We have a long message $m$ to encode. The message is composed of a set of symbols $\{s_i\}$. Let $p_i$ denote the number of appearance of $s_i$ in $m$. We seek to find a prefix-free code for each $s_i$...
Given a vector $v\in\mathbb Z^n$ and pairwise commutative matrices $M_1,\dotsc,M_k\in \operatorname{SL}(n,\mathbb Z)$, how to compute all 01-vectors in the orbit of $v$ with respect to multiplication ...