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Questions tagged [bipartite-matching]

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Matching Algorithm - How to maximize matched quantity with unique matching rules?

Given a set $S=\{A,B,\cdots,H\}$. Elements in $S$ can be matched according to the following rules: $$\begin{aligned} A\leftrightarrow B\\ C\leftrightarrow D\\ B+C\leftrightarrow F\\ D+A\...
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2answers
22 views

Minimum cost to match $n$ people with $m$ shops

We are given coordinates of $n$ people and $m$ shops. We should find a matching such that each person is matched with exactly one shop, and one shop is matched with at most one person.The total cost ...
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9 views

Online bipartite matching problem for task assignment

I have $n$ drivers, each one has a balance (in Us dollars), availability status (true if he is not working already) and number of accomplished tasks in the current ...
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2answers
37 views

Does real linear programming produce bipartite perfect matching using maxflow reduction?

Given a bipartite graph the standard reduction to max flow is with the construction similar to following diagram: We can formulate max flow as an linear programming problem with integer variables in ...
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1answer
15 views

Algorithm to assign producers to consumers with respect to connections

I am trying to analyze supply chains in a game and have come across this problem: First, an informal description: I have producers and consumers. Each producer produces a certain amount of goods, ...
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1answer
22 views

Hungarian Algorithm - Bipartite Graph Approach

I have been having some difficulty making sense of the Hungarian Algorithm outlined here. It seems incomplete and/or erroneous to me. The main issue is the line: If R_T ^ Z is nonempty, then ...
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21 views

Bipartite vertex cover [duplicate]

If this link can be any help https://codeforces.com/blog/entry/63164 A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. A ...
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2answers
39 views

Minimizing catastrophic risk in Gale-Shapley matching

In the hospital-resident assignment problem we have to match a large set of med students with a small set of hospitals. Hospitals may accept multiple students, but the number of students is much ...
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1answer
73 views

minimum cardinality maximal matching of graph

How to find minimum cardinality maximal matching? I tried that pick a edge from highest degree vertex remove other edges from same vertex and so on.
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1answer
38 views

Matching schedules between users and providers

I have a problem I've been dealing for the past few days, and I'm pretty stuck. Each user has a schedule for a given week, such as ...
4
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1answer
102 views

Finding a subset with few neighbors

Given a bipartite graph $G(X+Y,E)$, how can I find a non-empty subset $Y'\subseteq Y$, such that $|N(Y')| \leq |Y'|$ (where $N$ is the set of neighbors)? If $|Y|\geq |X|$ then the problem is easy - $...
2
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1answer
85 views

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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2answers
60 views

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 ...
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1answer
52 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 ...
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1answer
42 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 ...
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0answers
43 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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0answers
47 views

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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1answer
26 views

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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1answer
55 views

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 ...
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2answers
35 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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0answers
46 views

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 ...
2
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1answer
900 views

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 ...
2
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1answer
285 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 ...
4
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1answer
432 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, ...
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96 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 ...
5
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1answer
979 views

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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1answer
89 views

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 ...
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1answer
58 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 ...
2
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1answer
91 views

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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2answers
215 views

Find a minimum-cardinality Hall-violator

Given a bipartite graph $(X,Y,E)$, in which there is no perfect matching, I want to find a smallest subset that violates Hall's condition, i.e., a minimum-cardinality set $S \subseteq X$ for which $|...
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1answer
211 views

Weighted Matching with multiple assignments and min assignments

I need to do a weighted matching between two sets (say students and professors). The set of students is much larger than set of professors. So multiple students can be matched to professors. However, ...
2
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1answer
98 views

Variant of bipartite matching, with real capacities from source and to sink, all others unlimited

I've got a variant of bipartite graph matching and I can't find any literature about it. We have bipartite graph with real capacity edges from source to left vertices (the sum of which is 1), real ...
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1answer
190 views

Is Gale Shapley globally optimal?

Let us have a set of N men and N women, and we have two matrices of affinities $M$ and $W$ such that $M(i,j)$ is the affinity of the ith man towards the jth woman and $W(i,j)$ is the affinity of the ...
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0answers
68 views

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 ...
3
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1answer
684 views

What is the best algorithm to match a student's schedule with a tutor's schedule?

I am building an application (RoR framework) that can help to match a tutor and a student based on their subjects, budgets, locations and freetime. I have done the first three parts(subjects, budgets, ...
3
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1answer
174 views

3✕n chessboard with holes - maximum number of knights not attacking each other

I'm trying to to create an algorithm (working in polynomial time) to solve the following problem: What maximum number of knights that any two of them don't attack each other can be placed on a 3✕n ...
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643 views

Network Flow - Bipartite Matching: Doctors Without Weekends Problem

Problem You've periodically helped the medical consulting firm Doctors Without Weekends on various hospital scheduling issues, and they've just come to you with a new problem. For each of the next n ...
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1answer
616 views

maximum matching in a bipartite graph for solving a chess rook maximization problem

There's an n x n chessboard where some cells are instead holes. I want to have as many rooks as possible in a way that the rooks won't be able to capture each other....
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1answer
203 views

Given 2 sets of n points: minimize sum of traveled distances

I am given two sets $S, T$ each of $n$ points in $\mathbb{R}^k$, I want to find a bijection $a : S \rightarrow T$, such that $$\sum_{s \in S} d(s, a(s))$$ gets minimized, with $d$ being the Euclidean ...
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2answers
1k views

Reducing max flow to bipartite matching?

There's a famous and elegant reduction from the maximum bipartite matching problem to the max-flow problem: we create a network with a source node $s$, a terminal node $t$, and one node for each item ...
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1answer
319 views

Find perfect matching faster than MCM, in graph that has perfect matching?

Given an unweighted bipartite graph which has a perfect matching, is there an algorithm for finding a perfect matching in the graph that is faster than the best known algorithm for finding a maximum ...
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2answers
237 views

Auction where each bidder bids on a bundle of items

Is there some optimal solution in an auction where each bidder bids on a bundle of items?
5
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1answer
611 views

Hungarian Algorithm - Arbitrary Assignments

I've looked at several explanations of the Hungarian Algorithm for solving the Assignment Problem and the vast majority of these cover only very simplistic cases. The most understandable explanation ...
2
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1answer
477 views

Polynomial time solution for bipartite matching

Inspired by this StackOverflow question, I am wondering if there is an efficient algorithm for the following problem: Assume $n$ items and $n$ boxes, with all boxes numbered numerically and all ...
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0answers
512 views

Faster maximum weight matching algorithm in bipartite graph

I need to do a maximum weight matching in bipartite graphs rather than maximum weight perfect matching (which means that there is no need to match all the nodes). The nodes each side are both (at ...
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1answer
133 views

Problems that are easy on bipartite but hard on general graphs

Are there any problems that are easy for bipartite graphs, but hard for general graphs? I am asking because some classical problems are formulated in reference to people looking for a spouse, such as ...
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1answer
263 views

Why is bipartite graph matching hard?

I am reading on how solving maximum flow (Ford-Fulkerson) can be also used to solve unweighted bipartite graph matching problem. I think I don't understand the essence of this problem, because to me ...
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2answers
166 views

Minimizing the overall cost over groups

I am trying to solve the problem of minimizing the overall cost over several groups. The schema of the data goes something like this: ...
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1answer
629 views

Multiple matching in Maximum Flow problem?

I'm sorry if this has already been asked before, but I couldn't find any similar questions. The situation is as such: Assume there are x restaurants, each with a capacity q, and y people, each of ...
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0answers
156 views

Online bipartite edge-cover problem with requirements

I have $N$ nodes $v_1,\ldots,v_N$ in one partition $X$ and $M \leq N$ nodes $u_1,\ldots,u_M$ in a different partition $Y$. I want to connect nodes in $X$ to nodes in $Y$ with edges under the following ...