Questions tagged [bipartite-graph]
46 questions
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Name and complexity of this problem on bipartite graphs
Let $G=(U, V, E)$ be a biparite graph, with $U$ and $V$ being the two sets of nodes.
I am trying to find the smallest set of nodes $\hat{V} \subseteq V$ such that, for every node $u \in U$, $\hat{V}_u$ contains at least one element and is different…
Luigi Procopio
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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 $C \subseteq X$ we define: $$\sigma(C)=\min\{0,…
0410
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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 condition).
My goal is to find a subset…
AvidLearner
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Algorithm to find a set of nodes with a smaller set of neighbours in a bipartite graph
Given a bipartite graph, find a set of nodes on one side that has greater cardinality than the set of its neighbours on the other side.
This is a conceptually simple problem, but I suspect it is actually quite difficult.
Ray Butterworth
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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
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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 latter.
If we do not use integer variables…
Turbo
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Is every X3SAT instance with no cycles satisfiable?
Exactly 1 in 3 SAT (X3SAT) is a variation of the Boolean Satisfiability problem. Given a set of clauses, where each clause has three literals, is there an assignment such that in each clause exactly one literal is true? X3SAT is NP-hard even if we…
Russell Easterly
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Maximum cardinality b-matchings with the smallest number of bottom nodes
Given a bipartite graph G=(X,Y,E) and consider the b-matchings where each X-node is matched with at most one Y-node and each Y-node is matched with at most 2 X-nodes.
Is there a polynomial-time algorithm to find a b-matching with maximum cardinality…
Yohai Trabelsi
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Algorithm for maximum non-crossing edge set in bipartite graph with a fixed permutation
I'm trying to identify an algorithm to solve this computational problem
Input:
Bipartite graph (V, W, E), with E ⊆ V×W
A fixed order for both V and W: V = (v1, ..., vn) and W = (w1, ..., wm)
Output:
The subset of E of maximum cardinality in which…
Jordan
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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 $k \leq n$. How can we find the $k$ unique agents, who when…
user89692
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Finding a Maximum Cut With Force Labeled Vertices for Planar Graphs
The maximum cut problem is a combinatorial optimization problem that seeks to partition the vertices of a graph into two sets, $S$ and $T$, in a way that maximizes the number of edges that cross between the two sets. The problem is tractable for…
Daniel García
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Dinitz’ algorithm in simple unit-capacity networks
I am studying for an algorithm design course, and can't understand this demonstration about how Dinitz’ algorithm computes a maximum flow in $O(m \sqrt{n})$ time.
This is what is written on the slides I am reading:
Theorem. [Even–Tarjan 1975] In…
Placido Pellegriti
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Is it possible to have a 2 by 2 rigid framework without having a corresponding connected bipartite graph?
According to the theorem(see reference) on the rigidity of frameworks:
A rectangular framework is rigid if and only if its associated bipartite graph is connected.
Now consider the case for a 2-by-2 rectangular framework.
In this case, if we…
Aniruddha
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For a regular bipartite graph with vertices $X\cup Y$, prove that $|S|\leq|n(S)|$ $\forall S\subseteq X$
As the title states, we are given a bipartite undirected graph $G=(X\cup Y,E)$ such that every vertex $v\in V$ satisfies $d(v)=k$ for a constant $k$.
The general goal of the proof is to show that under these terms, the graph has a perfect matching…
Aishgadol
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Assignment Problem with Minimum and Maximum constraints
I have the following problem:
In a school, there are n students and m clubs, with n > m. Each student needs to be assigned a club. The students have preferences, (say top 3 or top 5) of the clubs they wish to be matched to. So far, it's basically…
devam_04
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