Questions tagged [bipartite-matching]

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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 ...
Yann's user avatar
  • 111
9 votes
2 answers
1k views

Size of Maximum Matching in Bipartite Graph

Am I correct in my observation that the cardinality of the maximum matching $M$ of a bipartite graph $G(U, V, E)$ is always equal to $\min(|U|, |V|)$?
ultrajohn's user avatar
  • 270
9 votes
3 answers
635 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 $|...
Y.Zhang's user avatar
  • 91
9 votes
2 answers
3k 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 ...
templatetypedef's user avatar
8 votes
0 answers
269 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 ...
D.M.'s user avatar
  • 81
7 votes
1 answer
960 views

Complexity of Hopcroft-Karp

I have a rather basic question about the number of operations taken by the Hopcroft-Karp algorithm for finding a maximum matching in a bipartite graph. It is commonly reported as $O(m \sqrt{n})$ where ...
Javin Aldrecht's user avatar
7 votes
1 answer
6k views

How to find the maximum independent set of a directed graph?

I'm trying to solve this problem. Problem: Given $n$ positive integers, your task is to select a maximum number of integers so that there are no two numbers $a, b$ in which $a$ is divisible by $b$...
palatok's user avatar
  • 255
7 votes
1 answer
263 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 ...
Dennis Lewis's user avatar
7 votes
1 answer
533 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 ...
user695652's user avatar
6 votes
2 answers
1k 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 ...
Cris's user avatar
  • 187
6 votes
1 answer
980 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 ...
Tom Baxter's user avatar
6 votes
1 answer
442 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 ...
Narek Bojikian's user avatar
5 votes
2 answers
9k views

The stable marriage algorithm with asymmetric arrays

I have a question about the stable marriage algorithm, for what I know it can only be used when I have arrays with the same number of elements for building the preference and the ranking matrices. ...
Little's user avatar
  • 163
5 votes
2 answers
183 views

How to match two point sets to minimize total distance?

Let's say we have two sets $X = \{x_1, \ldots, x_n\} \subset \mathbb R^d$, $Y =\{y_1,\ldots, y_n\} \subset \mathbb R^d$, how can we find a permutation $\pi$ such that $$D = \sum_{i=1}^n d(x_i, y_{\pi(...
flawr's user avatar
  • 373
5 votes
2 answers
35k 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 ...
Mahesha999's user avatar
  • 1,745
5 votes
3 answers
1k 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 ...
Dany's user avatar
  • 151
5 votes
2 answers
599 views

In the Hopcroft-Karp algorithm, what is the purpose of the breadth first search?

In the Hopcroft-Karp algorithm for bipartite matching, I don't understand the purpose of the breadth first search. I think it's used to find a set of vertex disjoint augmenting paths, but I'm not ...
Duncan's user avatar
  • 665
5 votes
1 answer
3k 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 ...
Banach Tarski's user avatar
5 votes
3 answers
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 ...
TestUser5's user avatar
5 votes
1 answer
183 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 ...
AvidLearner's user avatar
5 votes
2 answers
6k 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 ...
yjc's user avatar
  • 171
5 votes
1 answer
535 views

Decomposing a bipartite graph of maximal degree d to d matchings

I have tried for the last few days to prove that any bipartite graph of maximal degree d may be broken into (at most) d matchings. My main approach is to prove this inductively over the maximal ...
user3661799's user avatar
5 votes
2 answers
2k views

Bipartite Graph Game

So say we have a bipartite graph G=(X,Y,E). Let's make a game out of it. I go first. I pick a node in X. You go next. You pick a node in Y that is connected by an edge to the node I picked. Next it's ...
User's user avatar
  • 69
5 votes
1 answer
81 views

The maximum matching of a bipartite graph $(S, T)$ is $|X|+\min\limits_{A \subseteq X} (\min\{0, |N_G(A)|-|A|\}$, where $X \in \{S, T\}$?

Here is the full version of the problem I'm dealing with. Let $G=(S,T;E)$ be a bipartite graph and let $X$ be one of the two classes of its bipartition (i.e., $X \in \{S,T\}$). For a subset $C \...
0410's user avatar
  • 75
5 votes
2 answers
1k views

Maximum bipartite matching when some nodes must be matched

Consider the problem of finding a maximum cardinality bipartite matching under the additional condition that some set $S$ of nodes (all lying on the same side of the bipartition) must be matched. ...
Roberto Z.'s user avatar
5 votes
1 answer
276 views

Finding $k$ claws ($K_{1,3}$ bipartite graphs) in a graph?

Usually questions deal with claw-free graphs, but suppose we are given a graph $G$ and there are $k$ vertex-disjoing claws in the graph, how can we derive a randomised algorithm using color coding to ...
hysoftwareeng's user avatar
5 votes
0 answers
738 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 ...
Kaho Chan's user avatar
  • 161
4 votes
3 answers
2k views

Number of ways to fill a 2xN grid with M colors

This question was asked in the onsite regionals for ACM ICPC 2013 at Amritapuri. In short, the question asked to find the number of ways to fill a $ 2\times N$ grid with $M$ colors such that no two ...
Kyuubi's user avatar
  • 273
4 votes
1 answer
149 views

Bipartite graph question

Assume you are given a bipartite graph $G = (U, V, E)$ and you are given an integer $n$. Assume also that for each $v \in V$, you are given two integers $v_{min}$ and $v_{max}$ (where $v_{min} \le v_{...
Paul Reiners's user avatar
4 votes
1 answer
231 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 ...
Erel Segal-Halevi's user avatar
4 votes
1 answer
340 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 $V'$?
Farah Mind's user avatar
4 votes
1 answer
263 views

Term for a matching which is perfect on one side only

What is a standard term for a matching in a bipartite graph, in which one part has less vertices than the other part, and the part with less vertices is fully matched (but the other part is, obviously,...
Erel Segal-Halevi's user avatar
4 votes
1 answer
4k views

Stable marriage problem with only one side having preferences [duplicate]

I was wondering about a variation on the Stable Marriage Problem. Initially, we have two sets of entities, usually males and females, and they have preference lists ranking the other group, and ...
Vishaal Kalwani's user avatar
4 votes
2 answers
360 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 ...
Turbo's user avatar
  • 2,862
4 votes
1 answer
2k 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, ...
mathkid77's user avatar
4 votes
2 answers
269 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?
Mike's user avatar
  • 83
4 votes
1 answer
441 views

Maximum bipartite matching with extra reward for covering certain sets

Consider the following variation of Bipartite Maximum Matching. As usual, we have a bipartite graph $G$. In addition, there is an additional collection of sets $S_1,S_2,\dots,S_k$, with each set $...
jjohn's user avatar
  • 474
4 votes
1 answer
371 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\...
luca590's user avatar
  • 141
4 votes
1 answer
189 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 - $...
Erel Segal-Halevi's user avatar
4 votes
1 answer
261 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 ...
S.walia's user avatar
  • 41
4 votes
1 answer
133 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 ...
kangalio's user avatar
  • 141
4 votes
1 answer
435 views

Assignment problem with no cost

I have a problem that I was able to conceptualize as following: Problem We have a set of n people. And m subsets representing their ethnicity like White, Hispanic, Asian etc. Given any combination of ...
user2038833's user avatar
4 votes
1 answer
111 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 ...
Erel Segal-Halevi's user avatar
3 votes
1 answer
4k views

Find perfect matching whose weight is minimal, in polynomial time

Given a bipartite graph $G=(A,B,E)$ and a weight function $w: E \rightarrow\mathbb{R}^+$, I'd like to find a perfect matching $M\subseteq E$ with min. weight. I'm assuming $|A| \leq |B|$, and WLOG $G$ ...
Roman Vale's user avatar
3 votes
1 answer
264 views

Why is bipartite perfect matching a special case of clique problem?

In Lovász writes [1] : bipartite graph has a perfect matching, which is a special case of the clique problem Why is bipartite perfect matching a special case of clique problem? The Work of A.A. ...
user avatar
3 votes
1 answer
2k views

Number of Matchings in a Bipartite

Given two sets A and B of sizes |A| = n and |B| = m, where m >= n. There are edges from set A to set B. I need to find the number of matchings where all of vertices ...
Kyuubi's user avatar
  • 273
3 votes
1 answer
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 ...
Me.'s user avatar
  • 478
3 votes
1 answer
156 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 ...
apilat's user avatar
  • 131
3 votes
1 answer
64 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 ...
dysonsfrog's user avatar
3 votes
1 answer
35 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 ...
Adam Michael Wood's user avatar