# Questions tagged [bipartite-graphs]

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### Counterexample for the 1-optimal matching algorithm in Gabow's and Tarjan's paper on scaling algorithms for networks

Context I am reading Faster scaling algorithms for network problems by Gabow and Tarjan where I am researching part 2: "Matching and extensions". However, I am a bit confused with the ...
• 21
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### Weighted bipartite matching with no-cycle constraint

Given a weighted bipartite graph, I need to find a maximum-weight matching with the following additional constraint: the residual graph of the chosen matching is not allowed to contain any cycles. By ...
• 233
1 vote
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### Development details of the Hungarian algorithm for Maximum Perfect Bipartite Matching

There are two realization forms of Hungarian algorithm. One is the original dynamic matrix, and the other is via equality subgraph. I just checked the original paper of Hungarian method by Kuhn, which ...
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### Constrained Bipartite Matching

Let $G = (X,Y,E)$ be a bipartite graph. For some $A \subseteq X$ we say that $A$ can be perfectly matched if there is a matching $M \subseteq E$ such that all vertices in $A$ are matched; that is, for ...
• 412
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### Bipartite Matching with a Partition Constraint over the Vertices

Let $G = (X,Y,E)$ be a bipartite graph and let $X_1,\ldots,X_r$ be a partition of $X$. For some $i \in \{1,\ldots, r\}$ and $E' \subseteq E$ we say that constraint $X_i$ is covered by some $E'$ if ...
• 412
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### On a modular inverse graph construction

Given a balanced bipartite graph $G_1$ on $2n$ vertices on the condition $PM(G_1)\equiv1\bmod2$, an integer $i$ of size $\Omega(n\log n)$, can we find a balanced bipartite graph $G_2$ on $poly(n)$ ...
• 13.1k
140 views

### Error in TAOCP 4a on the bipartite graph constructed from a hypergraph

The first sentence on page 33 of Donald Knuth's The Art of Computer Programming (TAOCP) Vol. 4a reads: Furthermore, a hypergraph is equivalent to a bipartite graph with vertex set $V \cup E$ and ...
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### Complexity of maximum k-edge-colorable subgraph of a bipartite graph

Can the maximum $k$-edge-colorable subgraph of a bipartite graph be found in polynomial time? Equivalently, can the maximum $k$-colorable subgraph of the line graph of a bipartite graph be found in ...
• 7,590
145 views

### On perm+1 and det+1

Given a balanced bipartite graph G and a planar graph H. We do not know the number of perfect matchings in G and we do not know the number of spanning trees in H. But assume they are at least 3 both. ...
• 13.1k
138 views

### Complexity and Algorithm for specific Vertex Separator Problem

Given a graph $\Gamma=(V,E)$ with vertex set $V$ and edge set $E$ a $\textit{three partition}$ is decomposition of $V$ into a triple $(V_1, S, V_2)$ such that vertices of $V_1$ are only incident to ...
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### Is Finding an *Unbalanced* Biclique in Bipartite Graphs Hard?

In the balanced biclique in bipartite graphs (MBB) problem we are given a bipartite graph $G = (L,R,E), |L| = |R| = n$ and the goal is to find an induced subgraph of $G$, $G' = (L',R',E')$, with as ...
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1 vote
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1 vote
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1 vote
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### Bipartite graph projections, with threshold

Let $G=(\top,\bot,E)$ be a bipartite graph: $E\subseteq \top\times\bot$. The projections $G_\bot = (\bot,E_\bot)$ and $G_\top = (\top,E_\top)$ of $G$ are defined as follows: two vertices are linked ...
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1 vote
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### algorithms for a large submatrix / general factor / quasi-biclique problem?

Given a sparse 0/1 matrix $X$, too large to fit in memory, with $m$ rows and $n$ columns, I'm looking for an algorithm for finding a submatrix (when one exists) with maximum number of rows such that ...
• 145
1 vote
120 views

### Algorithm for maximum bipartite matching with arriving edges?

Given a bipartite graph with fixed nodes and incrementally arriving edges, is there any efficient algorithm to compute and update the maximum matching?
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### Is "two or zero" matching in a bipartite graph NP complete?

I have this problem, which I haven't seen before in the literature: given a bipartite graph and a natural number $k$, can we select at least $k$ of the edges such that each left-hand vertex is ...
3k views

### Complexity of counting matchings in a bipartite graph

I might be missing something obvious but I can't find references about the complexity of counting matchings (not perfect matchings) in bipartite graphs. Here is the formal problem: Input: a bipartite ...
• 9,697
690 views

### Bipartite Graphs - Maximum subset of one partition with at most n neighbours - NP-hard?

Given: A bipartite graph G=(U,V,E) Integers n and k. Decision Problem: Is there a subset of U of size k that has at most n neighbours? I am trying to figure out whether this problem is NP-hard (...
773 views

### Constrained version of vertex cover in a bipartite graph

Let $G(V_1, V_2, E)$ be a bipartite graph such that degree of all the vertices in $V_1$ is bounded by some constant (say) $d$. Now, for given two positive integer $l$ and $k$, we wish to decide if ...
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### Assignment problem with multiple workers for each job

I am wondering if there are any results on the following version of the assignment problem. We are given a set of jobs $J$ and a set of workers $W$, and for each job $j$ and worker $w$ we are given ...
342 views

### Sparser Bipartite graphs?

Maximal Planar Bipartite graphs are sparser than maximal planar graphs. For which other classes of graphs are maximal Bipartite members sparser than arbitrary maximal members. Let $\mathcal{C}$ be a ...
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I'm interested in a variant of the maximum weight matching in a graph, which I call "Maximum Fair Matching". Assume that the graph is full (i.e. $E=V\times V$), has even number of vertices, and that ...