Questions tagged [bipartite-matching]
The bipartite-matching tag has no usage guidance.
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Greedy Maximum Bipartite Matching
To find the maximum matching on a bipartite graph, I propose the following greedy algorithm:
At each iteration, pick an unmatched vertex with the smallest degree and match it to one of it's neighbours ...
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Expected maximum matching size in a random bipartite graphs
What is the expected maximum matching size of a bipartite graph $(A\cup B, V)$ where $\lvert A\rvert = n$ and $\lvert B\rvert = n$ and the probability of a edge existing between $A$ and $B$ is a fixed ...
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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 ...
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Find an assignment discarding a subset of possible assignments
We have a $N \times N$ cost matrix where the cost denotes the amount incurred for assigning a worker to a task.
The number of possible assignments is $N!$. Let us call this set of all possible ...
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A bipartite maxiumum matching problem, fitting a ratios vector on the resulting subset
Statement of problem
We have two sets of vertices, $V$ and $U$, each of which has a vector of attributes $A$.
The set of edges $E$ is defined such that there is an edge $vu\in{E}$ between vertices $v$ ...
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Schoolclass Optimization Algorithm for finding Stable Matching
I have the task to write a program that puts students in classes and that in the best possible way.
We have given the name, the foreign language a student chooses(french or latin), a profile (Music,...
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Bipartite matching with constraints on one part
We have a bipartite graph with parts $A$ and $B$, and it is edge weighted. We have some constraints for part $B$. Each constraint is in this format: Between vertices $b_1$ and $b_2$ both from part $B$,...
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What is the name of this matching problem?
We have a bipartite graph consisting of parts $A$ and $B$. Each vertex $i$ of part $A$ has weight $w_i$ and capacity $c_i$. We say a vertex $i$ in part $A$ is satisfied if at least $c_i$ adjacent ...
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Budgeted min cost max flow in bipartite where the flows must also be a matching set
I'm trying to find a problem description that is roughly akin to a budgeted min-cost max-weight bipartite matching where the capacities are greater than 1. Imagine a max-flow problem on a graph that ...
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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 ...
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A variation of the maximum bipartite matching problem
Given a bipartite simple graph $G=(V,E)$, where $V=A\cup B$ and $A\cap B=\emptyset$, any edge in $E$ connects two vertices in $A$ and $B$, respectively. The maximum bipartite matching problem is to ...
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Assign items from inventory to people maxmizing the number of satisfied people
We have a set of people, and each person has a list of wished items (not unique, they could want multiple copies of each item). We have an inventory of items that we want to assign to the people. We ...
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Maximum weighted matching in Bipartite Graph
I was solving a coding question which boiled down to this problem.
Given a bipartite graph $G=\{V\cup U,E\}$. There is a positive value given for every node in $U$.
Now we have to find the matching ...
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How to argue that an $A$-covering matching exists in this bipartite graph?
In lecture the following was mentioned in the context of matchings in bipartite graphs:
Let $U$ be a finite set and let $\mathcal{S}$ be a family of subsets of $U$.
For $u \in U$ let $r(u) := \lvert \...
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Matching students with companies based on their preference
I have a list of companies with n timeslots (number of slots may vary from company to company) and a list of students. Each student made a list of their top 3 companies they would like to talk to.
Is ...
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Proving that this matching is stable
Consider the stable marriage problem with $n$ men and $n$ women. Let $A$ and $B$ be two stable matchings, and suppose that we form a new matching $C$ by assigning to each men his favorite partner ...
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Matching problem in bipartite network with more than one edge per vertex
I'm interested to know if there is an algorithm to find possible solutions for the matching problem, in a bipartite network where each vertex have maximum number of connections greater than one. For ...
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Find a perfect matching with weights as close as possible to each other
Given a set of jobs $J$ and a set of machines $M$, where the link between machine $i\in M$ and job $j\in J$ has a positive weight $w_{ij}$. The problem is to select a perfect matching between the jobs ...
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Proving existence of sinkless orientation on graph with minimum degree 2
I am given a graph of minimum degree at least 2 (not necessairly regular). I want to prove that there is a way to orient the edges of G such that each node of G has at least one out-going edge.
As a ...
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confusion about Hopcroft-Karp time complexity analysis
From Wikipedia:
"Each phase increases the length of the shortest augmenting path by at least one: the phase finds a maximal set of augmenting paths of the given length, so any remaining ...
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Optimal prune of a mutually-exclusive bipartite graph
I start with a complete edge-weighted unbalanced bipartite graph.
For a known, fixed $n$ on the order of 1000:
$$0 \lt n, n \in I$$
Left and right cardinalities might not be equal; for sides $U$ and $...
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Using Flow graph to find maximum matching
I recently submitted an answer to the following question (homework in algorithms course):
A guy has m shirts, n pants, and p belts. he wants to make the maximum amount of outfits while abiding by ...
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Max-Min Weighted Matching
The maximum weighted matching problem (https://en.wikipedia.org/wiki/Maximum_weight_matching) finds a matching in a weighted graph that has maximum sum of weights. I was wondering if there are any ...
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Graph in which greedy algorithm for maximum matching is a 2-approximation
Here is a greedy algorithm for maximum bipartite matching:
Iteratively select an edge that is not incident to previously selected edges.
This algorithm returns a 2-approximation, and runs in linear ...
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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 \...
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Creating an algorithm which utilizes an already known optimal solution to max matching
Assume that there is a maximal matching of size k in an bipartite graph, G=(U,V,E). I now want to utilize this maximal matching in order to find a maximal matching in the bipartite graph where we add ...
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Hardness result for online matching
Currently studying the following paper:
"Fair Allocation in Online Markets" - Gollapudi and Panigrahi 2014
In which they present Theorem 2 as a hardness result for online maxmin matchings (...
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How to convert Bipartite Perfect Matching to SAT?
SAT is $NP$-complete while Bipartite Perfect Matching is in NC under derandomization assumptions. How to convert Bipartite Perfect Matching from balanced bipartites to SAT without Cook-Levin?
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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(...
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Is there a reduction from 2sat to bpm?
Given a 2SAT instance can we convert into bipartite perfect matching in parsimonious reduction?
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Bipartite Graph to solve the wolf river crossing problem
I have just started studying about bipartite graphs and there is an example that bipartite graph can be use to solve the wolf, cabbage, and the sheep river crossing problem, as a kid i had fun solving ...
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Assignment Problem with Minimum and Maximum constraints [duplicate]
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 ...
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Algorithms for alignment of posets
I have been looking for a while but I cannot find an algorithm for a particular generalisation of sequence alignment.
I have two posets $(A, <)$ and $(B, <)$ and a similarity score $s(a, b)$ for ...
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How to simulate online matching algorithms (implementation)
I was reading about online algorithms and bipartite matching.
I found an implementation that works fine on several websites (like geeksforgeeks).
For the online version, I found this paper
https://...
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Find approximate 'best' matching pairs by calculating the fewest possible weights
My specific problem is as follows:
Given two list of texts (in the order of 5 to 50 items)
Find best matching pairs with their corresponding matching score (weight)
Where each item can only be ...
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Predicate variant of Assignment Problem
Given two equally sized sets, $P$ of Boolean predicates and $E$, I want to decide if there exists a bijective function $f: P \rightarrow E$, such that
\begin{align}
\forall p \in P \; p(f(p))
\end{...
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why does relabel take O(VE) time total for unit capacity flow networks?
It is well known that for arbitrary flow networks, Goldberg's push-relabel algorithm takes $O(V^2E)$. Part of that comes from $O(V^2E)$ non-saturating pushes. Another part comes from $O(V)$ ...
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Flow graph with non zero lower bound or 0 capacity
I am afraid the question title might not be sufficiently accurate but I could not come up with something more accurate
Here is the problem
Given 'n' machines
Each machine has a set of capabilities
...
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Tight sets w.r.t. Hall's condition
Consider a bipartite graph G=(U+V,E) and suppose |U|=|V| and that G has a perfect matching. Therefore by P. Hall's condition, for every subsets A of U, the neighborhood N(A) of A has size at least |A|....
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Min weighted edge cover: don't follow proof in Schrijver
I'm reading section 19.3 of Combinatorial Optimization by Schrijver where he details an algorithm for finding the min-weight edge cover. His method works for general graphs, but I'm particularly ...
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Bipartite maximum matching with added constraints
Suppose you have two lists as follows
List $A$ = $(a_1, a_2, ..., a_m)$
List $B$ = $(b_1, b_2, ..., b_n)$
Each element in list $A$ can be paired with many or no elements in list $B$. I have a function ...
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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 ...
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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 ...
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Is there such a problem as b-Matching with different b values?
Consider a bipartite Garph $G=(L \cup R, E)$. Naturally, a b-Matching problem is to find a set of edges $M \subset E$, such that each node in $L$ and $R$ are adjuscent to maximum $b$ neighbors, and a ...
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Bipartite graphs with min weights
I have a full bipartite graph with node sets $A=\{a_1,a_2,\ldots,a_n\}$ and $B=\{b_1,b_2,\ldots,b_n\}$. Each node has a weight, $v_i$ for $a_i$ and $w_i$ for $b_i$. Each node $a_i$ is connected to ...
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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 ...
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Problem with understanding Multi-party security circuit for secure stable matching
I am reading the following paper:
MPCircuits: Optimized Circuit Generation for Secure Multi-Party Computation
Paper Link
I have following question:
We have two groups shown in the circuit. Why we ...
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Bipartite Graph in a Digraph
How do you find a sub-digraph in a digraph such that the in degree and out degree of each vertex is 1. My professor told in the class that an algorithm can be build for it using bipartite matching but ...
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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 ...
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What is the time complexity of the Edmonds-Karp algorithm for finding a maximum cardinality matching in bipartite graphs?
What is the time complexity of the Edmonds-Karp algorithm (not the Hopcroft-Karp algorithm) for finding a maximum cardinality matching in bipartite graphs? Is it still $O(|V||E|^2)$, or it has a ...
