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Questions tagged [bipartite-matching]

176 questions
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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 interesting problem to avoid throwaway computations…
Yann
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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 $|N(S)|<|S|$. This problem is the optimization…
Y.Zhang
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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 to be matched, then add appropriate…
templatetypedef
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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 $\min(|U|, |V|)$?
bettersayhello
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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 problem? I'm aware that perfect matching has a…
D.M.
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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 distance. I am aware that this transportation…
user695652
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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 I've found is a YouTube video. I can code the…
Tom Baxter
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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 $m$ is the number of edges in the graph and $n$…
Javin Aldrecht
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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 a movie with $h_1$. Then player 1 responds with…
Dennis Lewis
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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 $b$. I have to find the Maximum independent set and…
palatok
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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 theorem for weighted graphs as well. I tried…
Banach Tarski
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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. For example if we have 10 students that should be…
Little
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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 conected to some of the vertices from $U$. If for…
Cris
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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 came up with where solved on paper in almost…
Narek Bojikian
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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 most) the level of 10^3 and are usually unbalanced in…
Kaho Chan
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