All Questions
17 questions
0
votes
1
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77
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Bipartite matching where every adjacent pair of vertex on the left side of the graph has at least 1 vertex matched
The question is as stated above. I want to devise bipartite matching algorithm where it determies whether every adjacent pair of vertex on the left side of the bipartite graph has at least 1 vertex ...
2
votes
1
answer
315
views
Bipartite matching with a pairwise constraint
A long time ago I remember seeing a very clever construction for the following problem, but I can't find a reference for it anywhere: suppose I have a bipartite graph $G=(U\cup V, E)$, and the ...
- 4,049
4
votes
0
answers
118
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Reorganizational matching
Motivation. My friend works in an organization that is re-organizing itself in the following somewhat laborious way: There are $n$ people currently sitting on $n$ jobs in total (everyone has one job). ...
- 48.8k
2
votes
0
answers
163
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Generalizing Hall's marriage theorem to non-perfect matchings
Let $G = (X, Y, E)$ be bipartite graph such that $|X|=|Y|=n$.
A matching $M \subseteq E$ is a subset of disjoint edges
(i.e., there does not exist a pair of edges $(x, y) \in M$ and $(x', y') \in M$ ...
- 121
5
votes
1
answer
1k
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Bipartite graph with exactly one perfect matching
$\textbf{Problem:}$ Find all bipartite graphs $G[X,Y]$ satisfying the following properties:
$1.$ $|X|=|Y|$, where $|X|\ge 2$ and $|Y|\ge 2$.
$2.$ All vertices have degree three except for two vertices ...
- 181
1
vote
1
answer
312
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Unique bipartite perfect matchings and cycles?
Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G_{\pi,\pi'}$ be union of $G$ and $G_{\pi,\pi'}$ where $G_{\pi,\pi'}$ ...
- 13.9k
2
votes
2
answers
223
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Number of subgraphs with matching of size $n$ for a complete bipartite graph
Say we have a $K_{n,n}$ bipartite graph (i.e. a complete bipartite graph with $n$ nodes on each side). We induce a subgraph by deleting some subset of edges. There are $2^{n^2}$ possible subgraphs. ...
- 41
7
votes
1
answer
969
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Graph to Bipartite conversion preserving number of perfect matchings
Given a graph $G$ on $n$ vertices is there a technique to convert to a balanced bipartite graph $B$ with $O(n^c)$ vertices at some fixed $0<c$ in $O(n^{c'})$ time at some fixed $0<c'$ such that ...
- 13.9k
4
votes
2
answers
985
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Applications of Hafnians
I am learning about Hafnians but I am struggling to find real-world applications of them. I understand the applications of determinants, permanents, and even pfaffians but I am at a loss for Hafnians. ...
- 111
-1
votes
1
answer
378
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Marriages in infinite bipartite graphs with many neighbors
Let $A,B\neq \emptyset$ be disjoint and suppose $G = (A\cup B, E)$ is bipartite where for all $e\in E$ we have $e\cap A \neq \emptyset\neq e\cap B$. For $a\in A$ we set $N_G(a) = \{b\in B: (\exists e\...
- 48.8k
5
votes
2
answers
4k
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Solving assignment problem using Hungarian method vs min cost max flow problem
The traditional solution for the assignment problem is the Hungarian method - it's complexity is O(V^4) or O(V^3) if using Edmonds method.
However, it can also be reduced to a min cost max flow ...
- 201
4
votes
1
answer
1k
views
Polygamous stable marriage/ assignment problem
I'm not sure under which 'algorithm' it falls under, but here is the problem:
I need to match each person to 5 people from the opposite gender (each guy gets 5 girls, each girl gets 5 guys). Not all ...
- 201
6
votes
2
answers
7k
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How many perfect matchings in a regular bipartite graph?
We have a $d$-regular bipartite graph $G = (X,Y,E)$ with $|X| = |Y| = n$ and $|E| = nd$.
What is an upper bound on the number of perfect matchings of $G$?
- 61
0
votes
1
answer
773
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Counting matchings in a bipartite matching-covered graph
A graph is called matching-covered if every edge is containd in a perfect matching. (Such graphs are also sometimes called "elementary", e.g. in Chapter 4 of "Matching Theory" by Lovasz & Plummer)....
- 6,962
4
votes
0
answers
207
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Bounds on numbers of matchings of given sizes in bipartite graphs
I am interested in the following question:
For which sets ${m_1,\ldots m_k}$ of positive integers do there exist bipartite graphs having exactly $m_i$ matchings of size $i$ for each $1\leq i \leq k$, ...
- 1,327
13
votes
2
answers
811
views
Gale-Shapley stable marriage theorem: can we entrust matchmaking to monkeys?
Disclaimer: This is a question I have not done any real research about. I asked it myself some 5 years ago, and back then I had no idea where to start. Now I have some texts on stable matchings lying ...
- 33.8k
8
votes
1
answer
2k
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Condition on a bipartite graph to have an $m$-factor
This might be the most stupid question I am ever posting here: I am asking for a proof or a counterexample to a problem I proposed on MathLinks long ago.
Let $G$ be a bipartite graph, i. e., a graph ...
- 33.8k