# Questions tagged [bipartite-matching]

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### Assignment Problem — finding the $k$ agents with the best assignment

I have a question that I have been thinking about. Suppose we have $n$ agents, $m$ tasks, a cost matrix with $M_{ij}$ being the cost of agent $i$ performing task $j$, and are given a value $k \leq n$. ...
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### How to find maximum matching edges in undirected tree

Let $B$ be an undirected tree with $|V|$ nodes given as adjacency list. I want to develop a greedy algorithm using pseudo code to find a maximal matching in runtime $\mathcal{O}(|V|)$. My approach: ...
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### Computing minimum partition of poset of $N$ intervals into chains in $o(N^{2.5})$ time?

Consider a set $P$ of $N$ intervals $\{I_i = (l_i, r_i)\}$ partially ordered according the standard interval order: $I_i < I_j$ iff $r_i \le l_j$. I want to find a minimum cardinality partition of ...
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### An example where the algorithm of Hopcroft and Karp performs poorly?

I have been trying to construct an example, where Hopcroft and Karp's algorithm for the maximum matching problem performs poorly (say at least $\Omega(\log n)$ rounds). However, all the examples I ...
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### Bipartite graph minimal amount of vertices required

I have a bipartite graph made of two sets (SET 1 and SET 2) and I want to determine how many vertices from the ...
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### Maximum cardinality bipartite matching when nodes are ordered and only subsets can be matched?

Maximum bipartite matchine problem can be converted to the maximum flow problem and it can be solved by Edmonds-Karp algorithm in O(VE)<=O(V^3). But there can be bounded problem, when the nodes on ...
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### Using LP to prove the max matching - min cover theorem

Konig's theorem says that, in a bipartite graph, the size of the maximum matching equals the size of the minimum vertex cover. This theorem has several proofs; I would like to know if the following ...
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### Either find a perfect matching, or return a witness that none exist [duplicate]

I am looking for a polynomial-time algorithm that takes as input a bipartite graph $(X\cup Y, E)$, and returns one of two options: If a perfect matching exists, it returns the matching; Otherwise, it ...
104 views

### Multiple rounds of bipartite matching problem

I have a set of investors (say n), and a set of startups (say m). At the start, I have all the investors say either yes or no to the startup (which corresponds to whether they want to interact with ...
63 views

### More efficient maximum bipartite matching

I've been looking into weighted matching in bipartite graphs and am currently looking at maximum matchings in weighted bipartite graphs. As I've been reading and poking around at different books and ...
69 views

### Hungarian algorithm to search over all matching?

I am working on the following problem- "Finding the matching among all possible matching such that the sum of edge weight is minimum in the matching." Please note that I like to search over all ...
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### Find the set of edges in a bipartite graph such that the sum of edge weights is maximum satisfying some constraints

Let $G$ be a bipartite graph with sides $L$ and $R.$ Let $w_{lr}$ be the edge weight of an edge from $l \in L$ to $r \in R.$ Let $x_r$ be the node weight of the node $r \in R.$ Let $E$ denote the set ...
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### Determine whether two collections of items can be paired

Given collections I (items) and S (slots), where I >= S. And a pairing function that ...
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### How can I find matchings in a Bipartite graph beginning with specific vertices?

Context: I'm modelling kidney exchanges through directed acyclic graphs. I convert these to Bipartite graphs (by splitting each node into a donor and receiver, and the edge from the original graph ...
37 views

### Can a perfect matching always be found by a picking sequence?

There are $n$ people and $n$ items. For each person, there is a set of items he likes. Our goal is to give to each person a single item that he likes, i.e, find a perfect matching in the preference ...
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### Finding maximum bipartite matching

I read an article https://www.geeksforgeeks.org/maximum-bipartite-matching/ to solve the maximum bipartite problem. In this article, two solutions are given. In the first solution, they have ...
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### Perfect matching in a bipartite regular graph in linear time

Given a $G=(V,E)$ bipartite, undirected, 4-regular graph, I would like to find a perfect matching in linear time. It is easy to show that there is a perfect matching for the graph, by using flow and ...
355 views

### Bipartite Perfect Matching “Assignment Problem” - finding an assignment of a particular weight

The assignment problem is to find the minimum weight perfect matching in a weighted bipartite graph. This problem can be solved using the Hungarian algorithm in polynomial time. It is also possible to ...
638 views

### Changing preference in Gale-Shapley algorithm?

Suppose, in the context of the classic marriage problem, two equal size groups of $n$ men and $n$ women are being matched, with the GS algorithm. If a man were to switch the order of a pair of women, ...
113 views

### Complexity of removing edges to eliminate a perfect matching

Suppose $G$ is a bipartite graph which has a perfect matching. I want to find the fewest number of edges to delete from $G$ so that a perfect matching no longer exists. What is the complexity of this ...
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### Konig's Theorem for Min Weight Vertex Cover?

Koning's theorem states that the cardinality of the maximum matching in a bipartite graph is equal to the size of its minimum vertex cover. Wikipedia states that there is an equivalent version of the ...
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### I have n boys and n girls. I need to pair as much of them as possible for a dance in O(nlogn). Reduce this to a standard problem?

There are n girls and n boys. Each girl i has an objective attractiveness constant Pi (a natural number). The bigger the number, the more attractive. Each boy has a range in which he is comfortable ...
63 views

### Understanding characterizations of Matching on Graphs

I am studying Matching Theory on Graphs and I am wondering if I understand the characterization of the problems right. Definition: Let $G = (V, E)$ a graph. A set $M \subseteq E$ is called a matching ...
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### How to find n-1 complete matches for bipartite graphs that are related (speed-date)?

How can one assign n people pairwise to n-1 tables, in a speed-date fashion, such that no two persons meet twice and each person is at each table exactly once? Does this problem have a name? The ...
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### Find If a node exists in all maximum bipartite matchings

Given a bipartite graph, I need to find for each node, If this node exists in all the possible maximum matchings of the given graph or not. Note that there can be multiple maximum matchings of a ...