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1answer
34 views

Union grouping in bipartite graphs?

I'm trying to figure out a good (and fast) solution to the following problem: I have two roles I'm working with, let's call them players and teams having many-to-many relationship (a player can be on ...
1
vote
1answer
71 views

Stable marriage problem with only one side having preferences

I was wondering about a variation on the Stable Marriage Problem. Initially, we have two sets of entities, usually males and females, and they have preference lists ranking the other group, and ...
3
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1answer
57 views

Finding a subset in bipartite graph violating Hall's condition

We are given a bipartite graph of $n \leq 200$ vertices in both the first and the second partite set. Let $U$ be some set of vertices in the first set, and $V$ those vertices from the second that are ...
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0answers
77 views

Finding the number of distinct maximal matching in a bipartite graph [closed]

In a bipartite graph, how can we find the total number of ways of getting a maximal matching? The cardinality of both the sets in the bipartite graph may not be the same. So two matchings are said to ...
4
votes
1answer
150 views

Complexity of Hopcroft-Karp

I have a rather basic question about the number of operations taken by the Hopcroft-Karp algorithm for finding a maximum matching in a bipartite graph. It is commonly reported as $O(m \sqrt{n})$ where ...
4
votes
1answer
43 views

Decomposing a bipartite graph of maximal degree d to d matchings

I have tried for the last few days to prove that any bipartite graph of maximal degree d may be broken into (at most) d matchings. My main approach is to prove this inductively over the maximal ...
2
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1answer
58 views

Bipartite Matching in the Plane

I'm currently working on a problem that I came across: You are given a set $B$ of $n$ points in the plane, and a set $R$ of $n$ points in the plane. Each point is given by its coordinates. I have ...
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2answers
120 views

Concrete and simple applications for bipartite graphs [closed]

I am looking for concrete and simple problems that may be solved using bipartite graphs or bipartite graph properties. Any idea along with explanations are welcome.
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0answers
46 views

Marriage algorithm that maximizes number of pairings

I have a bipartite graph similar to the marriage problem, where there are M males and N females, and a 1:1 matching between males and females is desired (with the remainder of the more populous gender ...
2
votes
1answer
36 views

Determining the minimum vertex cover in a bipartite graph from a maximum flow/matching using the residual network rather than alternating paths

Wikipedia shows how one can determine the minimum vertex cover in a bipartite graph ($G(X \cup Y, E)$) in polytime from a maximum flow using alternating paths. However, I read that the (S,T) cut ...
1
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1answer
22 views

The min cut capacity in a network based on a bipartite graph (Hall's Theorem)

Thanks to Yuval Filmus, I got to read these lecture notes by Trevisan. At the bottom half of Page 5, The capacity of cut $S$ is the number of edges that go from $S$ to $\overline{S}$, that is ...
2
votes
1answer
58 views

How to optimally seperate a student body?

Students will identify certain students they want to work with. I have therefore decided to split them into two groups where I want to minimize the number of people in Group 1 who want to work with ...
1
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1answer
57 views

maximum bipartite matching

I am working out with the rooks problem. If there are m rooks on an nxn chessboard,i have to give describe a polynomial (in m and n) time algorithm that finds a maximum-sized subset of the rooks such ...
5
votes
3answers
194 views

Maximum number of matched vertexes in a one-to-many bipartite graph

I have a variant of bidding problem at hand. There are N bidders(~20) who bid for items from a pool of many items(~10K). Each bidder can bid many items. I want to maximize the number of bidders who ...
1
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0answers
31 views

How to maximize the number of buyers in a shop?

There is a shop which consists of N items and there are M buyers. Each buyer wants to buy a specific set of items. However, the cost of all transactions is same irrespective of the number of items ...
1
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1answer
74 views

Local search: Problem with neighborhood definition

I have question on understanding the following neighborhood relation within a local-search approximation scheme. Let $M$ be a legal matching on any bipartite graph. Let $U_k$ be the neighborhood ...
0
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0answers
135 views

Hopcroft–Karp algorithm time complexity

In the last 2 paragraphs of the paper about Hopcroft–Karp algorithm to find the maximum cardinality matching in bipartite graph: https://dl.dropboxusercontent.com/u/64823035/04569670.pdf The ...
3
votes
3answers
347 views

Number of ways to fill a 2xN grid with M colors

This question was asked in the onsite regionals for ACM ICPC 2013 at Amritapuri. In short, the question asked to find the number of ways to fill a $ 2\times N$ grid with $M$ colors such that no two ...
2
votes
1answer
59 views

Number of Matchings in a Bipartite

Given two sets A and B of sizes |A| = n and |B| = m, where m >= n. There are edges from set A to set B. I need to find the number of matchings where all of vertices ...
4
votes
2answers
140 views

Bipartite Graph Game

So say we have a bipartite graph G=(X,Y,E). Let's make a game out of it. I go first. I pick a node in X. You go next. You pick a node in Y that is connected by an edge to the node I picked. Next it's ...
3
votes
1answer
207 views

In the Hopcroft-Karp algorithm, what is the purpose of the breadth first search?

In the Hopcroft-Karp algorithm for bipartite matching, I don't understand the purpose of the breadth first search. I think it's used to find a set of vertex disjoint augmenting paths, but I'm not ...
2
votes
1answer
408 views

Existence of bipartite perfect matching

Let $B = G(L, R, E)$ be a bipartite graph. I want to find out whether this graph has a perfect matching. One way to test whether this graph has a perfect matching is Hall's Marriage Theorem, but it is ...
3
votes
1answer
245 views

Assignment problem with no cost

I have a problem that I was able to conceptualize as following: Problem We have a set of n people. And m subsets representing their ethnicity like White, Hispanic, Asian etc. Given any combination ...
5
votes
1answer
1k views

How to find the maximum independent set of a directed graph?

I'm trying to solve this problem. Problem: Given $n$ positive integers, your task is to select a maximum number of integers so that there are no two numbers $a, b$ in which $a$ is divisible by ...
4
votes
1answer
90 views

Bipartite graph question

Assume you are given a bipartite graph $G = (U, V, E)$ and you are given an integer $n$. Assume also that for each $v \in V$, you are given two integers $v_{min}$ and $v_{max}$ (where $v_{min} \le ...
0
votes
1answer
87 views

Graph Bipartiteness

I have 2 questions regarding Bipartiteness with corresponding examples. 1) Can a non-connected graph be bipartite if it has an isolated vertex? Let's take the following graph: I would say YES with ...
1
vote
2answers
411 views

The stable marriage algorithm with asymmetric arrays

I have a question about the stable marriage algorithm, for what I know it can only be used when I have arrays with the same number of elements for building the preference and the ranking matrices. ...
2
votes
1answer
511 views

Clarification with Kuhn-Munkres/Hungarian Algorithm

I have been attempting to get my mind around the Kuhn-Munkres/Hungarian Algorithm. I have been using the following statement of the algorithm which I found here. Based on my readings on the ...
2
votes
1answer
147 views

Matching girls with boys without mutual attraction (variant of maximum bipartite matching)

Let us say you have a group of guys and and a group of girls. Each girl is either attracted to a guy or not, and vice versa. You want to match as many people as possible to a partner they like. Does ...
8
votes
2answers
257 views

Size of Maximum Matching in Bipartite Graph

Am I correct in my observation that the cardinality of the maximum matching $M$ of a bipartite graph $G(U, V, E)$ is always equal to $\min(|U|, |V|)$?