Questions tagged [bipartite-matching]

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1answer
50 views

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 $k \leq n$. ...
1
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1answer
64 views

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: ...
4
votes
1answer
70 views

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 ...
3
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1answer
23 views

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 ...
2
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2answers
51 views

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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0answers
11 views

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 ...
3
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0answers
87 views

What is the exact time complexity of randomized Kuhn's algorithm?

Please, read the whole question before answering, the exact details of the implementation are important. Suppose that you want to find largest cardinality bipartite matching in bipartite graph with $...
0
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0answers
26 views

Counting step in Gale-Shapley Algorithm(Deferred Acceptance Algorithm)

Imagine there is a modified version of many-to-one school choice matching with Deferred Acceptance Algorithm(DAA): other things will be the same as original DAA, except that for an unassigned student, ...
11
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1answer
321 views

How fast can we compute the size of maximum matching in an unweighted bipartite graph?

Is there a way to compute the size of a maximum matching in an unweighted bipartite graph more efficiently (e.g. faster) than computing a maximum matching? It is a long shot but it is often an ...
1
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1answer
24 views

Maximum-cardinality matching in unbalanced bipartite graphs

Let $G = (X+Y, E)$ be a bipartite graph, and suppose we want to find a maximum-cardinality matching in $G$. The Hopcroft-Karp algorithm runs in time $O(|E|\sqrt{|V|})$, where here $|V| = |X|+|Y|$. So ...
1
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1answer
32 views

Student Course Allocation Problem with Many Constraints [closed]

Problem statement In an university, there are $t$ course categories, $m$ courses, $n$ sections, $p$ students. $i$-th section has: A student capacity: $cap_i$. Two lecture timings. (Formally, each ...
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0answers
9 views

How to assign people to categories based on some voting rules?

Say in an election there are predefined categories users can be assigned to. Of the people who are assigned to a certain category, other users can vote for them. To compute the results, the following ...
3
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1answer
2k 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
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0answers
44 views

Alternative criterion for approximate maximum-weight perfect matching algorithms [closed]

Is there any literature on approximate maximum-weight perfect matchings where the approximation criterion is not the factor between the approximate and exact weight sum achieved by each solution, but ...
1
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0answers
13 views

bipartite d regular expender explicit construction

I am looking for an explicit (and simple) construction of a d regular bi bipartite graph which is an expander. I searched the web and didn't find any sufficient answer. The only explicit graph I did ...
4
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1answer
79 views

Term for a graph decomposition based on a maximum matching

Let $M$ be a maximum cardinality matching in a bipartite graph $G(X+Y,E)$. Let $X_0$ be the subset of $X$ unmatched by $M$. Define the following sequence: $Y_1 = $ the neighbors of $X_0$ using edges ...
4
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1answer
142 views

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\...
7
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2answers
253 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 $|...
1
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2answers
30 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 ...
4
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1answer
71 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 ...
4
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2answers
88 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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0answers
19 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 ...
1
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1answer
20 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, ...
1
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1answer
24 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 ...
0
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0answers
26 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 ...
5
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3answers
918 views

Vertex cover in bipartite graph from Hopcroft-Karp Algorithm

Vertex cover in bipartite graph is polynomial algorithm: by König's theorem the number of edges in a maximum matching is the number of vertices in a minimum vertex cover. I've implementated the ...
3
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2answers
44 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 ...
5
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3answers
1k 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 ...
0
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1answer
129 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.
4
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1answer
107 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 - $...
5
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2answers
462 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 ...
2
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1answer
156 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 ...
1
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1answer
65 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 ...
1
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2answers
68 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 ...
1
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1answer
99 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 ...
0
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1answer
61 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 ...
1
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0answers
68 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 ...
2
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1answer
351 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 ...
2
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0answers
50 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 ...
3
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1answer
27 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 ...
1
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1answer
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 ...
2
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2answers
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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0answers
52 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 ...
4
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2answers
3k views

Finding a minimum weight perfect matching in Christofides TSP algorithm

Context: After creating the minimum spanning tree, the next step in Christofides' TSP algorithm is to find all the N vertices with odd degree and find a minimum weight perfect matching for these odd ...
2
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1answer
1k 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 ...
4
votes
1answer
636 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, ...
8
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0answers
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 ...
5
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1answer
1k 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 ...
1
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1answer
90 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 ...
2
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1answer
99 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 ...