All Questions
Tagged with combinatorial-optimization graph-theory
134 questions
5
votes
2
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557
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What is the proper name for this "tersest path" problem in Infinite Craft?
The web game Infinite Craft gives you a starting set of elements $V_0\subset V$ and a mapping $E$ of type $V\times V\rightarrow V$. In fact, $F$ is commutative: $E(v_a,v_b) = E(v_b,v_a)$. So another ...
CommunityBot
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1
vote
1
answer
86
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Complexity of calculating the optimal amalgamation of an optimal cycle-cover
Let $G(V,E)$ be a complete symmetric graph with positive edge weights and let further $\mathcal{C}=\lbrace C_1,\,\cdots\,C_k\rbrace$ be the minimum-weight vertex-disjoint cycle cover.
The set $E$ of ...
CommunityBot
- 1
5
votes
1
answer
383
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Shortest polygonal chain with $6$ edges visiting all the vertices of a cube
I am trying to find which is the minimum total Euclidean length of all the edges of a minimum-link polygonal chain joining the $8$ vertices of a given cube, located in the Euclidean space. In detail, ...
CommunityBot
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0
votes
0
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24
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Minimizing intersections between spanning trees of graph embeddings in polynomial time
Assume I have $N$ complete graphs $G_1, G_2,...,G_N$, and consider their embeddings $E_1, E_2,...,E_N$ in $\mathbb{R}^2$. Is there a (potentially stochastic) polynomial time algorithm to construct ...
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2
votes
1
answer
112
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Finding survivable paths with a set of vulnerable edges
Consider a graph $G=(V,E)$ and a source-destination pair $(s,t)$. A set of edges $E'\subseteq E$ are vulnerable in the sense that at most $k$ of them may fail. My problem is to find a set of $(s,t)$ ...
CommunityBot
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0
votes
1
answer
51
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Cycle-Sculpturing with Minimal Vertex-Deletion
given a simple, finite and symmetric graph $G(V,E)$ with $n$ vertices and at least $n$ edges
Question:
how can the smallest set of vertices $W\subset V$ be calculated for which the graph induced by $...
- 5,429
0
votes
1
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79
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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; ...
CommunityBot
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2
votes
1
answer
329
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Worst case performance of heuristic for the non-Eulerian windy postman problem
The windy postman problem seeks the cheapest tour in a complete undirected graph, that traverses each edge at least once; the cost of traversing an edge is positive and may depend on the direction, in ...
CommunityBot
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1
vote
0
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54
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Finding a path of given length with maximal relative weight
Let $G$ be a directed graph with vertices $V$ and edges $E \subset V\times V$. A path of length $n \geq 2$ in $G$ is a sequence of vertices $(i_{0},i_{1},\ldots,i_{n-1})$ such that $(i_{k},i_{k+1}) \...
- 798
1
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0
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48
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How to solve a optimal matching problem where the "quality" of matching is only determined once all nodes are matched
Note: I'm not a mathematician; I'm just a biologist with basic math background trying to solve a scientific problem, so please excuse my ignorance.
The gist of the problem is as follows:
I have two ...
- 5,429
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 := \...
- 63.9k
1
vote
0
answers
67
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Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows
From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
- 1,451
3
votes
2
answers
336
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Algorithm to evaluate "connectedness" of a binary matrix
I have the following problem: given an $m \times n$ binary matrix $A$ like e.g. the following $9 \times 39$ matrix:
...
- 5,429
0
votes
0
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123
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A variant of Steiner tree
Consider a directed Steiner tree problem with a source node $s$ a set $T$ of terminals with the following constraint. For each node $v$ on the tree, we assign a branching value $b_v$ as follows. (1) $...
- 367
8
votes
1
answer
567
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Joining the $2^k$ points of $\{0,1\}^k$ with the shortest tree
Let $k$ be a given positive integer, and then consider the unit hypercube $\{0, 1\}^k \subset \mathbb{R}^k$ (i.e., a $k$-dimensional "cube" in the well-known Euclidean space).
We need to ...
- 156k
0
votes
2
answers
139
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Graph vertices selection for paths sum minimalization
Let $G = (V, E)$ be a simple, finite, weighted, undirected, connected graph. Let $P = \{p_{ij}, p_{kl}, p_{il}, ... \}$ be a set of paths, where $p_{ij}$ is a path from $i$-th to $j$-th vertex. Is ...
1
vote
0
answers
88
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Is there an efficient algorithm for finding a fundamental cycle basis of a graph with the fewest odd cycles? Failing that, a hardness result on this?
I can think of a greedy algorithm:
Let $B$ be a fundamental cycle basis of graph $G$ induced by spanning tree (or forest) $T$
For $e\in T$, let $n_+(e)$ ($n_-(e)$) be the number of even (odd) cycles ...
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4
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0
answers
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) \...
CommunityBot
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1
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0
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157
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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
2
votes
1
answer
152
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First known proof of the $2 \cdot n-2$ Theorem for the planar generalization of the Nine dots problem
Reading the Wikipedia page about the well-know Nine dots puzzle, I have just seen that the planar generalization of this problem would have been proven in 1956 (see Wikipedia: Nine dots puzzle), while ...
- 188k
2
votes
2
answers
112
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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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0
votes
0
answers
16
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Complexity of finding single source paths with capacity constraints and length constraints
Let $G=(V,A)$ be a directed graph with distinguished vertex $s\in V$ and let $c:A\rightarrow{\mathbb N}$ denote arc capacities. For any $t\in V,t\not=s$ we are given two numbers: $C_{t},L_{t}$. Let $...
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11
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3
answers
409
views
Two disjoint trees
Let $G$ be a graph and let $A_1, A_2 \subseteq V(G)$ be disjoint sets of vertices. Let us call $(A_1, A_2)$ independent if there exist vertex-disjoint trees $T_1, T_2 \subseteq G$ within $G$ which ...
- 32.1k
2
votes
1
answer
825
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Partition of a graph into subgraphs with small maximum degree
Denote the maximum degree of an undirected graph $G=(V,E)$ by $\Delta$. That is, $\Delta = \max_{v\in V} \deg(v)$. I'm looking for a way to divide $G$ into two separate induced subgraphs $G_1=(V_1,...
- 32.1k
5
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1
answer
291
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Minimum number of edges to remove to have low degree
I have the following problem, where $k$ is a fixed integer.
Input: Graph $G$.
Output: Minimum number of edges to remove from $G$ to obtain a graph such that every node has degree at most $k$.
Do ...
- 32.1k
-2
votes
1
answer
174
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What is known about iterated matching as a TSP heuristic
A fairly wellknown heuristic for TSP that is based on matching is described in the 2003 paper Match twice and stitch: a new TSP tour construction heuristic by Andrew B. Kahng and Sherief Reda.
Its ...
- 13.2k
6
votes
1
answer
374
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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, ...
- 5,409
7
votes
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 ...
CommunityBot
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1
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0
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How to embed an arbitrary graph into (k,d)-kautz space (like multidimensional scaling of non-normed space)? See details in the following
How to embed an arbitrary graph into (k,d)-kautz space (like multidimensional scaling of non-normed space)? See details in the following.
Given a graph $G = \{V,E\}$,
we have a distance matrix (the ...
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12
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0
answers
14k
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A New York Times tiles-based graph theory question
The New York Times has a daily puzzle named Tiles that works as follows. Start with $m$ squares (in the official version, this is 30, in a 6x5 grid), and a set of $p>4$ possible patterns (typically ...
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0
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126
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On the absolute difference of the Laplacian eigenvalues of an unbalanced signed graph and its underlying graph
Let $\Sigma=(G,\sigma)$ be an unbalanced signed graph with the underlying connected graph $G=(V,E)$ and $\sigma:E\rightarrow \{-1,1\}$, the signing function. Let the Laplacian eigenvalues of $\Sigma$ ...
- 61
0
votes
0
answers
57
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Maximum number of h-dimensional hyper-edges without forming any (h+1) complete subgraph
This is about graph theory.
Define an h-dimensional hyperedge as a set that contains h vertices.
A graph of (h+1) vertices is h-complete if any h combination (or any subset with size h) is an h-...
- 12.9k
1
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0
answers
63
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A multi-layer version of Menger's theorem
Menger's theorem says that the maximum number of pairwise disjoint paths between two vertex sets $L,R$ of a graph G equals the minimum size of an $L$-$R$ separator. Below is a generalisation with more ...
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3
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0
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111
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The matrix representation of an interval graph
It is well-known that many classes of graphs have matrix representations that can be written concisely. For example,
The set of all directed acyclic graphs consists of binary matrices $x_{ij} \in \{...
- 4,049
3
votes
1
answer
240
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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 ...
- 32.1k
6
votes
1
answer
538
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Induced matching number
Definition:
A matching in a graph $G$ is a subgraph consisting of pairwise disjoint edges. If the subgraph is an induced subgraph, the matching is an induced matching. The largest size of an induced ...
- 32.1k
3
votes
0
answers
190
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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 ...
0
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0
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118
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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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1
vote
1
answer
536
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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 ...
- 1,451
5
votes
1
answer
429
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Menger's theorem with restrictions on where the paths can begin and end
Let $k\in\mathbb N$. Given a finite graph with two subsets of vertices $X$ and $Y$, Menger's Theorem gives a criterion for when there are $k$ pairwise disjoint paths starting in $X$ and ending in $Y$.
...
- 32.1k
0
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0
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50
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Can we talk about approximation when the decision problem for solution existence is NP-Hard
I am wishing to design an approximation algorithm for an optimization problem where the existence of solution for corresponding decision problem is not guaranteed. Is it wise to find an approximation ...
- 6,367
1
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0
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65
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Find a cut of a graph that minimizes the ratio between the edge weights of the cut and the edge weights inside one subgraph
Given an edge-weighted undirected graph $G=(V,E)$ (can assume the weights are non-negative) and a source node $v_s\in V$, a cut is a partition of $G$'s vertices into two complementary sets $S$ and $T$....
- 143
0
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1
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182
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Relationship between cycle length, number of chords, and number of induced $P_{4}$ subgraphs of the cycle
I was wondering if there was a known relationship between the length of cycle, the number of chords of the cycle, and the number of induced $P_{4}$ subgraphs of the cycle.
Here, I am referring to ...
CommunityBot
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2
votes
1
answer
255
views
Can connectivity be less than min cut/degree?
Suppose we have a graph with min-cut $\lambda$ and minimum degree $> \lambda$.
Is it possible for there to be a vertex that is at most $\lambda$-connected to every other vertex in the graph?
...
- 4,219
1
vote
2
answers
107
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Minimum edge-weighted directed subgraph in polynomial time
I am looking for an algorithm with polynomial complexity where, given a strongly connected edge-weighted digraph I can find the minimal subgraph which connects some root vertex v to a known set of ...
- 5,429
4
votes
2
answers
315
views
Connecting $2n$ points in $\mathbb R^2$ with line segments s.t. each point belongs to exactly one line segment
I'm trying to do a certain simulation related to the toric code and I'm looking for an algorithm that connects $2n$ points ($n \in \mathbb Z_+$) in $\mathbb R^2$ with line segments with the following ...
- 5,429
2
votes
0
answers
63
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Maximize connectivity probability with a number of edges
We are given a graph $G$, whose edges are either open or closed. Initially all the edges are closed. For each edge $e$, if we choose to activate it, then after the activation, it becomes open with ...
- 367
0
votes
0
answers
40
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Subtour-gluing constraints for ILP formulation of TSPs
If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-...
- 13.2k
1
vote
0
answers
45
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Vertex cover via maximally unbalanced spanning trees
The vertex cover problem asks for a smallest subset $U\subseteq V$ that is adjacent to all edges of a symmetric graph $G(V,E)$.
Inspired by the observation that led to this question Perfectly balanced ...
- 13.2k
1
vote
1
answer
165
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Combinatorial graph optimization problem on integer adjacency matrices
We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers.
Let $z_{i,j}:=\frac{M_{i,j}}{M_{i,j}+\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and
$z:=\...
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