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# Maximum matching bipartite algorithm

4-2 Lecture 4: Matching Algorithms for Bipartite Graphs Figure 4.1: A matching on a bipartite graph. P, as it is alternating and it starts and ends with a free vertex, must be odd length and must have one edge more in its subset of unmatched edges (PnM) than in its subset of matched edges (P \M). For example Maximum Bipartite Matching Maximum Bipartite Matching Given a bipartite graph G = (A [B;E), nd an S A B that is a matching and is as large as possible. Notes: We're given A and B so we don't have to nd them. S is a perfect matching if every vertex is matched. Maximum is not the same as maximal: greedy will get to maximal The bipartite matching is a set of edges in a graph is chosen in such a way, that no two edges in that set will share an endpoint. The maximum matching is matching the maximum number of edges. When the maximum match is found, we cannot add another edge. If one edge is added to the maximum matched.

The maximum matching of a graph is a matching with the maximum number of edges. This is very difficult problem. We study only maximum matching in a bipartite graph. In a bipartite graph the vertices can be partition into two disjoint sets V and U, such that all the edges of the graph have one end point vertex in U and the other end in V According to my understanding, the algorithm tries to find an augmenting path for each left vertex u (from 0 to n1-1). If such a path exists, u can be added to the matching, keeping all previously added vertices in the matching AALG5: Flow networks, maximum bipartite matching example Unweighted Bipartite Matching Decision 1 (D1) - Matchings - Bipartite Graphs and Maximum Matching Algorithm - Duration: 41:30

A maximum bipartite matching is a maximum matching on a digraph G which is bipartite. Given that G is bipartite, the problem of finding a maximum bipartite matching can be transformed into a maximum flow problem solvable with the Edmonds-Karp algorithm and then the maximum bipartite matching can be recovered from the solution to the maximum. A maximum matching (also known as maximum-cardinality matching) is a matching that contains the largest possible number of edges. There may be many maximum matchings. The matching number of a graph is the size of a maximum matching. Note that every maximum matching is maximal, but not every maximal matching is a maximum matching Maximum matching in bipartite and non-bipartite graphs Lecturer: Uri Zwick December 2009 1 The maximum matching problem Let G= (V;E) be an undirected graph. A set M Eis a matching if no two edges in M have a common vertex. A vertex vis matched by Mif it is contained is an edge of M, and unmatched otherwise A common bipartite graph matching algorithm is the Hungarian maximum matching algorithm, which finds a maximum matching by finding augmenting paths. More formally, the algorithm works by attempting to build off of the current matching, $$M$$, aiming to find a larger matching via augmenting paths Their algorithm is based on using a push-relabel maximum flow algorithm and then, when the matching created by this algorithm becomes close to optimum, switching to the Hopcroft-Karp method. Several authors have performed experimental comparisons of bipartite matching algorithms

### Maximum Bipartite Matching - tutorialspoint

• CS105 Maximum Matching Winter 2005 6 Maximum Matching Consider an undirected graph G = (V;E). Deﬁnition: A matching, M, of G is a subset of the edges E, such that no vertex in V is incident to more that one edge in M. Intuitively we can say that no two edges in M have a common vertex
• g relaxations, and primal-dual algorithm design. 1 Bipartite maximum matching In this section we introduce the bipartite maximum matching problem, present a na ve algorithm with O(mn) running time, and then present and analyze an algorithm due to Hopcroft and Karp that improves the running time to O(m p n). 1.1 De nitions De nition 1
• This video is a tutorial on an inroduction to Bipartite Graphs/Matching for Decision 1 Math A-Level. Please make yourself revision notes while watching this and attempt my examples. Complete the.

I write bipartite matching using dinitz's algorithm.Also there is a theorem that for the graphs of the type of the maximum bipartite matching problems it has the same complexity as relabel to front(and it is way easier to implement) 1. Lecture notes on bipartite matching February 9th, 2009 2 1.1 Maximum cardinality matching problem Before describing an algorithm for solving the maximum cardinality matching problem, one would like to be able to prove optimality of a matching (without reference to any algorithm) /***** * Compilation: javac BipartiteMatching.java * Execution: java BipartiteMatching V1 V2 E * Dependencies: BipartiteX.java * * Find a maximum cardinality matching (and minimum cardinality vertex cover) * in a bipartite graph using the alternating path algorithm

### Maximum Matching in Bipartite Grap

1. A perfect matching is a matching in which each node has exactly one edge incident on it. One possible way of nding out if a given bipartite graph has a perfect matching is to use the above algorithm to nd the maximum matching and checking if the size of the matching equals the number of nodes in each partition
2. Python Algorithm - Maximum Bipartite Matching - Graph Algorithm - A matching in a Bipartite Graph is a set of the edges chosen in such a way A matching in a Bipartite Graph is a set of the edges chosen in such a way that no two edges share an endpoint
3. The Hungarian matching algorithm, also called the Kuhn-Munkres algorithm, is a $$O\big(|V|^3\big)$$ algorithm that can be used to find maximum-weight matchings in bipartite graphs, which is sometimes called the assignment problem. A bipartite graph can easily be represented by an adjacency matrix, where the weights of edges are the entries.
4. Flow algorithm to ﬁnd Maximum-Sizematchings in bi-partite graphs. In this section we discuss how to ﬁnd Maximum-Weightmatchings in bipartite graphs, a sit-uation in which Max-Flow is no longer applicable. The O(|V |3)algorithm presented is the Hungarian Al-gorithm due to Kuhn & Munkres. • Review of Max-Bipartite Matching
5. A Scaling Algorithm for Maximum Weight Matching in Bipartite Graphs Ran Duan University of Michigan Hsin-Hao Su University of Michigan Abstract Given a weighted bipartite graph, the maximum weight matching (MWM) problem is to nd a set of vertex-disjoint edges with maximum weight. We present a new scaling al-gorithm that runs in O(m

Algorithms and data structures source codes on Java and Maximum matching for bipartite graph. Kuhn's algorithm in O(E*V) Maximum matching for bipartite graph. 1 Bipartite maximum matching In this section we introduce the bipartite maximum matching problem, present a na ve algorithm with O(mn) running time, and then present and analyze an algorithm due to Hopcroft and Karp that improves the running time to O(m p n). 1.1 De nitions De nition 1 Maximum Bipartite Matching Robin Visser De nition Example Network Flow Approach Construction De nition Algorithm Time Complexity Alternate Approach Algorithm Example Pseudocode Problem Examples Network Flow Approach We can solve the maximum bipartite matching problem using a network ow approach. We rst ensure that all edges from U to V are.

### bipartite - Augmenting Path Algorithm - Maximum Matching

In a maximum matching, if any edge is added to it, it is no longer a matching. There can be more than one maximum matching for a given Bipartite Graph. We have discussed importance of maximum matching and Ford Fulkerson Based approach for maximal Bipartite Matching in previous post. Time complexity of the Ford Fulkerson based algorithm is O(V x E) An $O(n^3)$ min-cost bipartite matching algorithm can be found in the public version of the U of T ACM codebook. I don't know whether it's the same as the Hungarian algorithm (which is $O(n^3)$) but it should suit your purposes number (nM) to every perfect matching, and thus does not change the problem. The goal in the minimum-cost perfect bipartite matching problem is to compute the perfect matching M that minimizes P e2M c e. The feasible solutions to the problem are the perfect matchings of G. An equivalent problem is the maximum-weight perfect bipartite matching

Algorithms and data structures source codes on Java and C++. Maximum matching for bipartite graph. Hopcroft-Karp algorithm in O(E * sqrt(V)) - Algorithms and Data Structure Maximum Matching in Bipartite Graphs . In many situations we are faced with a problem of pairing elements of two sets. The traditional example is boys and girls for a dance, but you can easily think of more serious applications Solve the Eatcake practice problem in Algorithms on HackerEarth and improve your programming skills in Graphs - Maximum Bipartite Matching 26.3 Maximum bipartite matching 26.3-1. Run the Ford-Fulkerson algorithm on the flow network in Figure 26.8 (c) and show the residual network after each flow augmentation This thesis applies two algorithms to the maximum and minimum weighted bipartite matching problems. In such matching problems, the maximization and minimization problems are essentially same in that one can be trans-formed into the other by replacing the weight on each edge with an inverse of the weight

### AALG5: Flow networks, maximum bipartite matching exampl

(2016) Expected time complexity of the auction algorithm and the push relabel algorithm for maximum bipartite matching on random graphs. Random Structures & Algorithms 48 :2, 384-395. (2016) Pairing for resource sharing in cellular device-to-device underlays Multithreaded Algorithms for Maximum Matching in Bipartite Graphs Ariful Azad1, Mahantesh Halappanavar2, Siva Rajamanickam 3, Erik G. Boman , Arif Khan 1, and Alex Pothen , E-mail: {aazad,khan58,apothen}@purdue.edu, mahantesh.halappanavar@pnnl.gov, and {srajama,egboman}@sandia.go A matching in a graph is a set of edges such that each vertex appears in at most one edge of . Try to find a maximum matching: a matching using as many edges as possible. Click an edge to select or unselect it CMPSCI611: The Bipartite Matching Problem Lecture 6 We saw last week that the greedy algorithm can fail to ﬁnd the maximum-weight matching in an arbitrary graph. In fact it can fail for the simpler problem of ﬁnding

### A Tutorial to Bipartite Matching Algorithm Topta

• imum among all perfect matchings •Maximum matching Not necessarily perfec
• Simple Maximum Bipartite Matching. So, you may have just learned this or similar augmenting path proof for finding the maximum cardinality matching in a bipartite graph! This is pretty much the direct translation of that proof into an algorithm. This is the augmenting path algorithm. Instead of converting it into a flow problem, this runs in O.
• Maximum matching in bipartite and non-bipartite graphs (Draft) Lecturer: Uri Zwick June 5, 2013 1 The maximum matching problem Let G= (V;E) be an undirected graph. A set M Eis a matching if no two edges in M have a common vertex. A vertex vis matched by Mif it is contained is an edge of M, and unmatched otherwise
• imum-cost matching in G. First, given a constant 2(0;1=3), we present, in Section 3, a deter

CS4245 Analysis of Algorithms Bipartite Matching. Istvan Simon. The Marriage Problem and Matchings . Suppose that in a group of n single women and n single men who desire to get married, each participant indicates who among the opposite sex would be acceptable as a potential spouse 2. Lecture notes on non-bipartite matching February 18th, 2009 6 (and this may result in further shrinkings) and when the algorithm terminates, we use Theorem 2.2 to expand it to a maximum matching in the original graph. Correctness. Now suppose that none of these possibilities apply any more for any of the Even vertices Try to find a maximum matching only for the nodes which are defined by the elements of X If it is a maximum matching(I.e all elements in the right side are matched) Add all the other nodes not already present and continue the augmenting path algorihtm Pick the maximum for all

A Matching in a graph G = (V, E) is a subset M of E edges in G such that no two of which meet at a common vertex.Maximum Cardinality Matching (MCM) problem is a Graph Matching problem where we seek a matching M that contains the largest possible number of edges Maximum Matching in Bipartite Graphs. The new algorithm works perfectly for any graph, provided there are no cycles of odd node count. In other words, the graph must be bipartite. Bipartite graphs work so well, in fact, that they will often terminate with a maximum matching after a greedy match Maximum Weight Bipartite Graph Matching 1 Introduction In this lecture we will discuss the Hungarian algorithm to ﬁnd a matching of maximum possible weight in a bipartite graph. We also discuss the integer programming formulation of the problem and its relaxation to Linear Programming(LP) problem. 2 Algorithm Consider a bipartite graph G = (V,E) I was wondering if there exists a way to compute the size of the maximum matching in a bipartite graph without having to compute the maximum matching itself. Any help or direction will be reall 26-6 The Hopcroft-Karp bipartite matching algorithm. In this problem, we describe a faster algorithm, due to Hopcroft and Karp, for $p$ finding a maximum matching in.

augmentation-based parallel algorithms for bipartite cardinality matching that demonstrates good speedups on multithreaded shared memory multiprocessors. I. INTRODUCTION We design, implement, and evaluate ﬁve parallel algorithms for computing a matching of maximum cardinality in a bipar-tite graph on multicore and massively multithreaded. Bipartite Matching Algorithm. Extension of a Ford Fulkerson max flow problem using depth first search. Problem Statement. Given an undirected bipartite graph, find the maximum matching between left & right halves. Each node in the Left half L mapped to at most 1 node in R. Similarly, each node in the Right Half R is mapped to at most 1 node in Good algorithms for maximum weighted matching in general graphs have been known for decades. Although free software implementations of these algorithms are available on the web, they are not so easy to find. This web page presents a new implementation of maximum weighted matching in Python and in Perl C++ Algorithm - Maximum Bipartite Matching - Graph Algorithm - A matching in a Bipartite Graph is a set of the edges chosen in such a way A matching in a Bipartite Graph is a set of the edges chosen in such a way that no two edges share an endpoint The maximum fractional matching problem is a relaxation of the maximum matching. The two values can di er in general, but for bipartite graphs they are equal. We include a proof for completeness. Claim 1 In a bipartite weighted graph, there exist a maximum matching with weight equal to the weight of the maximum fractional matching

A maximum matching is a matching of maximum size. The (same) solution: In general, the problem of finding a maximum matching in a graph is NP-hard, but if the graph is bipartite there is a polynomial solution. The problem above can be modelled with a bipartite graph Algorithm for bipartite graphs Let X, Ybe the bipartition of the vertices. Begin with the empty matching. Direct all edges from Xto Y. while ∃free vertex xin Xd CSC 373 - Algorithm Design, Analysis, and Complexity Summer 2016 Lalla Mouatadid Network Flows: Bipartite Matching We conclude our discussion of network ows with an application to bipartite matching. We need the following de nitions: A graph G(V;E) is a bipartite graph if V can be partitioned into two sets A and B, such that A[B = V Maximum Bipartite Matching. A Bipartite Graph is a graph whose vertices can be divided into two independent sets L and R such that every edge (u, v) either connect a vertex from L to R or a vertex from R to L Advanced algorithms build upon basic ones and use new ideas. So the big problem we're going to try and solve is known as bipartite matching. the maximum. Also, thinking of a way to bridge from @Jukka Suomela's solution to the algorithm from the question, there is a difficulty where we go from the maximum matching to the minimum vertex cover: while finding the maximum matching can be done by a max-flow-like algorithm, how do you recover the minimum vertex cover from it using a flow-like algorithm Lecture notes on non-bipartite matching 4 stem and shrink the blossom. We recursively nd a maximum matching in this graph G=B (and this may result in further shrinkings) and when the algorithm terminates, we use Theorem 2 to expand it to a maximum matching in the original graph. Correctness Abstract. Two parallel algorithms for finding maximum bipartite matchings on a CRCW PRAM are presented. The first algorithm finds a maximum bipartite matching in O(n log n) time using m processors, where n is the number of vertices and m is the number of edges in the bipartite graph CS294-5: Great Algorithms February 1, 2006 Professor Richard Karp Scribe: Brian Green Edmonds's Non-Bipartite Matching Algorithm Deﬁnition 1 (matching) A matching in a graph is a set of edges, no two of which meet at a common vertex. A maximum matching is a matching of maximum cardinality. 2 3 1 7 4 6 5 8 Figure 1. The three solid edges.

### Matching (graph theory) - Wikipedi

In which we show how to solve the maximum matching problem and the minimum vertex cover problem in bipartite graphs. Lecture 14 In this lecture we show applications of the theory of (and of algorithms for) the maximum ow problem to the design of algorithms for problems in bipartite graphs CS 3510 Design & Analysis of Algorithms Section A, Lecture #16 Bipartite Matching Instructor: Richard Peng Nov 1, 2017 DISCLAIMER: These notes are not necessarily an accurate representation of what I said during the class. They are mostly what I intend to say, and have not been carefully edited. Focus today is a special case of maximum In contrast to some other matching algorithms, the graph need not be bipartite. The algorithm was introduced by Jack Edmonds in 1965 and has been further improved since then. Many exact modern algorithms for the maximum matching problem on general graphs are still based on the ideas of the Blossom Algorithm Matching¶ Provides functions for computing a maximum cardinality matching in a bipartite graph. If you don't care about the particular implementation of the maximum matching algorithm, simply use the maximum_matching(). If you do care, you can import one of the named maximum matching algorithms directly

### Matching Algorithms (Graph Theory) - Brillian

1. Maximum flow and bipartite matching. Aug 20, 2015. The maximum flow problem involves finding a flow through a network connecting a source to a sink node which is also the maximum possible. Applications of this problem are manifold from network circulation to traffic control
2. You will be given two Bipartite Graphs, each with N vertices. The vertices in both the graphs are labeled from 1 to N.. You have to find the maximum size of a set S of vertices, such that it forms an Independent Set in the first graph, and the subgraph induced by S in the second graph has a perfect matching
3. • Max weighted matching in bipartite graphs. • Max weighted matching in general graphs. What we will discuss here are the ﬁrst two types of the problem. B. Approach for Solving Maximum Matching To solve the maximum matching problem, we need an algorithm to ﬁnd these maximum matching. The main ide Let W(M) = P (i,j)∈M wi,j denote the weight of a matching. We present three algorithms for maximum weighted bipartite matching. 3.1 First Algorithm: Negative cycles The ﬁrst algorithm is an extension of the algorithm for maxim um cardinality bipartite matching (ﬁgure 1) Distributed-Memory Algorithms for Maximum Cardinality Matching in Bipartite Graphs Ariful Azad, Aydın Buluc¸ E-mail: azad@lbl.gov, abuluc@lbl.gov Computational Research Division Lawrence Berkeley National Laboratory Abstract—We design and implement scalable distributed-memory algorithms for maximum cardinality matching in bi-partite graphs

### Hopcroft-Karp algorithm - Wikipedi

1. The push-relabel algorithm for bipartite matching. In this section, we describe the underlying sequential push-relabel algorithm for the bipartite matching problem. It is based on a simplification by Cherkassky et al. of the push-relabel algorithm for maximum flow introduced by Goldberg and Tarjan
2. imum (cardinality) vertex cover in a bipartite graph. A bipartite graph in a graph whose vertices can be partitioned into two disjoint sets such that every edge has one endpoint in either set
3. Repeat Repeat Repeat Answer Overall algorithm Time analysis Improvement Weighted Bipartite Graph Weighted Matching Maximum Weighted Matching Augmenting Path (change of definition) Idea for finding maximum weight matching Overall Algorithm Time analysis Stable Marriage Problem Stable Marriage Problem Stable
4. imum-cost matching. Hopcroft and Karp show that if the weight of each edge is 1, then a maximum cardinality matching can be.

upon the same outputs. A maximum matching algorithm is one that ﬁnds a match with the maximum size or weight. A maximum bipartite matching algorithm will give maxi-mum utilization of all output ports. As mentioned, a MSM is not necessarily desirable, since it can lead to instability and unfairness under admissible trafﬁc patterns, and starva The problem of finding a maximum matching in a graph is solvable in polynomial time in general graphs, but it has a very simple algorithm in bipartite graphs, that we shall see shortly. (The algorithm for general graphs is beautiful but rather complicated.) The algorithm is based on a reduction to the maximum flow problem

### Bipartite Graphs/Matching (Intro)-Tutorial 12 D1 Edexcel

Network flow algorithms find the maximum number of units we can send through the edges of this graph from the source to the sink, without exceeding the maximum capacity on any edge. It may not seen obvious, but if we have an algorithm to solve the network flow problem, we can find a bipartite matching for any graph There are M job applicants and N jobs. Each applicant has a subset of jobs that he/she is interseted in. Each job opening can only accept one applicant and a job applicant can be appointed for only one job It is a long-standing open question whether this algorithm can be derandomized. In this article, we give an almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs. This gives us a deterministic parallel (quasi-NC) algorithm for the bipartite perfect matching problem

Oblivious Maximum Bipartite Matching Size Algorithm with Applications to Secure Fingerprint Identiﬁcation Marina Blanton and Siddharth Saraph Department of Computer Science and Engineering University of Notre Dame mblanton@nd.edu, ssaraph@nd.edu Abstract. The increasing availability and use of biometric data leads to situa Maximum Bipartite Matching : Matching : A matching is a set of edges with no endpoints in common (Thus a matching in G(V,E) induces a pairing of some nodes in V using the edges in E.) In Bipartite Graphs, Matching is a set of edges, with no endpoints in common, with one endpoint in set M and the other in set Given an unweighted bipartite graph G = (V, E) V = (A, B), the maximum cardinality bipartite matching problem is to find a matching with maximum cardinality. Algorithm. Algorithms to solve this problem try to extend existing matchings by finding augmenting paths. When there are no more augmenting paths, the matching is a maximum matching  ### Fast Maximum Matching Algorithm for Bipartite Graph

algorithm for on-line weighted bipartite matching (with triangle inequality) must perform as badly as (2n 1) in arbitrary metric spaces. An interesting feature of our algorithm is that as each man arrives, it computes the best o -line maximum matching among the women and the currently arrived men to determine th On-line Bipartite Matching Made Simple Benjamin Birnbaum Claire Mathieuy Abstract We examine the classic on-line bipartite matching problem studied by Karp, Vazirani, and Vazirani  and provide a simple proof of their result that the Ranking algorithm for this problem achieves a competitive ratio of 1 1=e. Introductio In this lecture •Concepts of matchings •Hopcroft-Karp's algorithm for maximum cardinality matching in bipartite graphs •Edmonds's algorithm for maximum cardinalit

### BipartiteMatching.java - Algorithms, 4th Edition by Robert ..

maximum matching in a bipartite graph with restrictions. (See  for the properties of graphs and matchings.) This problem is shown to be NP-complete, and offers an expla- nation why matching approaches to scheduling are unsuccessful. If no restrictions ar 4.2 Bipartite Maximum Cardinality Matchings We will rst consider bipartite graphs, for which it is considerably easier to nd matchings. So assume in this section that Gis bipartite, with partite sets Aand B. We will focus on the maximum weight matching problem. One can reduce the maximum cardinality matching problem to a maximum ow problem, as. efficient algorithms by surveying algo- rithms for the four closely related problems of finding a maximum cardinality or weighted matching in general or bipartite graphs. Mathematically, these are all spe- cial cases of the problem of weighted matching in general graphs. However, w That is, when Haxell's condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find. (a) Draw a bipartite graph to show the trips that the drivers would like to do. Initially Mihi and Pat are assigned to Learnalott Senior School, Robert is assigned to Brayknee Junior School and Tony is assigned to Adupgood Senior School. (b) Starting from this matching use the maximum matching algorithm to find a complete matching. Yo

### Python Algorithm - Maximum Bipartite Matching - Graph

Maximum Cardinality Matching in Bipartite Graphs Maximum Cardinality Matchings and Node Covers in Graphs. A matching in a graph G=(V,E) is a subset M of the edges E such that no two edges in M share a common end node. A maximum cardinality matching is matching with a maximum number of edges 3.1 Bipartite matching. In the introductory chapters it turned out that finding a maximum cardinality matching in a bipartite graph plays a crucial role for assignment problems. Therefore, we discuss in this chapter various efficient methods for finding maximum matchings in bipartite graphs We design, implement, and evaluate GPU-based algorithms for the maximum cardinality matching problem in bipartite graphs. Such algorithms have a variety of applications in computer science. Maximum Bipartite Matching using Ford Fulkerson Algorithm (using DFS) - Jav bipartite matching. It is easy to see that the one-round communication complexity also gives a lower bound on the space needed by a one-pass streaming algorithm to compute a(1 )-approximate bipartite matching. The focus of this work is to understand one-round communication complexity and one-pass streaming complexity of maximum bipartite matching

A Parallel Tree Grafting Algorithm for Maximum Cardinality Matching in Bipartite Graphs Ariful Azad 1, Aydın Buluc¸ , and Alex Pothen2, E-mail: azad@lbl.gov, abuluc@lbl.gov, and apothen@purdue.ed The difference between our cycle cancelling algorithm and the one used for the general minimum cost flow problem is that we need to run a maximum flow algorithm to derive an initial network (here we simply assign the vertices to each other as we are guaranteed a perfect matching it's a complete bipartite graph) Building the matching is the bottleneck in the algorithm, the rest takes only linear time. The implementation from tryalgo.bipartite_matching import max_bipartite_matching def alternate (u, bigraph, visitU, visitV, matchV): extend alternating tree from free vertex u. visitU, visitV marks all vertices covered by the tree Relations between the maximum matching, minimum vertex cover, maximum independent set, and maximum vertex biclique for a bipartite graph 5 How to induce maximum matching and minimum vertex cover from each other in a bipartite graph Therefore, bipartite matching can be computed with Ford-Fulkerson in O(VE). Wrapup We have just seen an example of problem reduction : reducing the maximum bipartite matching problem to a flow problem and using a flow algorithm to solve it

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