All Questions
10 questions
2
votes
0
answers
163
views
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
2
votes
2
answers
223
views
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
views
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
3
votes
0
answers
75
views
Fraction of graphs with bound on number of perfect matchings
Asymptotically what is the fraction of balanced bipartite graph on $2n$ vertices with at most $cn^{\beta}$ edges having at most $n^\alpha$ perfect matchings for any fixed $c,\alpha>0$ and fixed $\...
- 13.9k
2
votes
0
answers
361
views
On symmetric difference of $k$-partite perfect matchings
Given a bipartite graph we know that symmetric difference of any two perfect matchings is union of even cycles.
Conversely when is it true that every union of even cycles comes from symmetric ...
- 13.9k
5
votes
2
answers
4k
views
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
views
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
views
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
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