Questions tagged [bipartite-matching]

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Problem with understanding Multi-party security circuit for secure stable matching

I am reading the following paper: MPCircuits: Optimized Circuit Generation for Secure Multi-Party Computation Paper Link I have following question: We have two groups shown in the circuit. Why we ...
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1answer
47 views

Bipartite maximum matching with added constraints

Suppose you have two lists as follows List $A$ = $(a_1, a_2, ..., a_m)$ List $B$ = $(b_1, b_2, ..., b_n)$ Each element in list $A$ can be paired with many or no elements in list $B$. I have a function ...
3
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1answer
25 views

Minimize range of distances between two sets of points

I have two sets of n points each in 2D Cartesian coordinates. I want to find a one-to-one pairing between the points in sets A ...
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1answer
28 views

Bipartite Graph in a Digraph

How do you find a sub-digraph in a digraph such that the in degree and out degree of each vertex is 1. My professor told in the class that an algorithm can be build for it using bipartite matching but ...
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1answer
53 views

Find a maximum matching that saturates a given set of vertices

In an unweighted bipartite graph $G = (X + Y,E)$, we would like to find a maximum matching, but among all those maximum matchings, we would like to find one that saturates a given subset $X_0\subseteq ...
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1answer
36 views

Is there such a problem as b-Matching with different b values?

Consider a bipartite Garph $G=(L \cup R, E)$. Naturally, a b-Matching problem is to find a set of edges $M \subset E$, such that each node in $L$ and $R$ are adjuscent to maximum $b$ neighbors, and a ...
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45 views

Why did Hopcroft and Karp write $M_0, M_1, M_2, \cdots, M_i, \cdots$? (Hopcroft - Karp Algorithm)

I am reading “An $n^{\frac{5}{2}}$ Algorithm for Maximum Matchings in Bipartite Graphs” by Hopcroft and Karp. Please see the image below. Let $s$ be the cardinality of a maximum matching. I think any ...
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2answers
61 views

Bipartite graphs with min weights

I have a full bipartite graph with node sets $A=\{a_1,a_2,\ldots,a_n\}$ and $B=\{b_1,b_2,\ldots,b_n\}$. Each node has a weight, $v_i$ for $a_i$ and $w_i$ for $b_i$. Each node $a_i$ is connected to ...
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1answer
54 views

Efficient algorithm to map two differently-sized sets of numbers as closely as possible?

The problem I have two sets of numbers and need to find a mapping between those two sets, so that the total distance between two mapped numbers is as small as possible. Two numbers must not be mapped ...
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2answers
28k views

Perfect matching in a graph and complete matching in bipartite graph

When I google for complete matching, first link points to perfect matching on wolfram. It defines perfect matching as follows: A perfect matching of a graph is a matching (i.e., an independent ...
5
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1answer
121 views

Saturated sets in bipartite graph

Let $G=(X\cup Y, E)$ be an unweighted bipartite graph. We are given that for every $W\subseteq X$ it holds that $|W|\leq |N(W)|$, where $N(W)$ is the neighborhod of $W$ in $Y$ (aka Hall's marriage ...
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33 views

What is the time complexity of the Edmonds-Karp algorithm for finding a maximum cardinality matching in bipartite graphs?

What is the time complexity of the Edmonds-Karp algorithm (not the Hopcroft-Karp algorithm) for finding a maximum cardinality matching in bipartite graphs? Is it still $O(|V||E|^2)$, or it has a ...
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56 views

Maximum Matching for Line Graphs

I tried to study the maximum matching for a line-graph , i.e , $L(G)$ of a graph , $G(V,E)$ , from here : https://www.sciencedirect.com/science/article/pii/S0012365X97001039 But could not understand ...
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58 views

How to solve this problem using the Maximum matching algorithm for general graph?

For any general graph G(V,E) , the maximum matching can be calculated in O(√V.|E|) time using the following algorithm : https://www.researchgate.net/publication/...
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1answer
43 views

Perfect Matching in Bipartite Graph with mutually exclusive edges

Problem I would to solve Perfect Matching in Bipartite Graph Problem where some edges are mutually exclusive. Example Left vertices: $a,b,c$ Right vertices: $x,y,z$ Edges: $(a,x),(a,y),(b,z),(c,y)...
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1answer
222 views

Maximum matching in a bipartite graph

Given a bipartite graph $G=(V_1 \cup V_2, E)$ and a set $V' \in (V_1 \cup V_2)$. What is the complexity of finding a maximum matching in $G$ that uses only $x$ vertices from $V'$?
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1answer
153 views

Hardness of a scheduling/assignment problem

I am trying to prove the hardness of the following problem. This problem is from google hashcode, qualification-round, 2020. Hier is a brief description of the problem. Given a list or libraries and ...
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1answer
308 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: ...
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1answer
181 views

Winning strategy for a given game on graphs

The game goes as follows. Two players are playing a game, player 1 and player 2, in which the first player starts by naming a hero $h_1$, then player 2 responds with a villain $v_1$ who has played in ...
3
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1answer
107 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 $k \leq n$. ...
4
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1answer
97 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 ...
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1answer
33 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 ...
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2answers
83 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
145 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 $...
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1answer
382 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 ...
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1answer
72 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 ...
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1answer
43 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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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
60 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 ...
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0answers
19 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
87 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
230 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\...
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2answers
333 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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2answers
239 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
112 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
163 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
34 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
35 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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0answers
29 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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3answers
999 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
45 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 ...
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1answer
262 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
128 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
620 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
351 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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1answer
80 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 ...
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2answers
86 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 ...
2
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1answer
198 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
84 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 ...