Questions tagged [bipartite-matching]

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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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2 votes
2 answers
35 views

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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1 vote
1 answer
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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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2 votes
0 answers
29 views

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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2 votes
1 answer
58 views

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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1 vote
1 answer
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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 $...
1 vote
1 answer
30 views

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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1 vote
1 answer
28 views

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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1 vote
0 answers
100 views

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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5 votes
1 answer
65 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 \...
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1 vote
1 answer
24 views

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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Find a matching of a bipartite graph that uses all vertices in the left set

I have a bipartite graph already divided into a left and right set, and I would like to find a matching that uses all of the vertices in the left set. To phrase this problem less abstractly, I have a ...
1 vote
0 answers
70 views

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 (...
1 vote
1 answer
138 views

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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5 votes
2 answers
103 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(...
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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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1 vote
2 answers
207 views

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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2 votes
0 answers
106 views

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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1 vote
0 answers
51 views

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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1 answer
49 views

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://...
1 vote
0 answers
37 views

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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1 vote
1 answer
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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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1 answer
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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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1 vote
1 answer
63 views

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 ...
2 votes
0 answers
34 views

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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2 votes
1 answer
151 views

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 ...
1 vote
1 answer
156 views

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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3 votes
1 answer
74 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 ...
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4 votes
1 answer
147 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 ...
2 votes
1 answer
309 views

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 ...
1 vote
2 answers
98 views

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 ...
4 votes
1 answer
101 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 ...
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1 answer
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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 ...
0 votes
1 answer
52 views

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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5 votes
1 answer
154 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 ...
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2 votes
0 answers
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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 ...
0 votes
0 answers
58 views

Maximum Matching for Line Graphs

I tried to study the maximum matching for a line-graph , i.e , $L(G)$ of a graph , $G(V,E)$ , from here : https://www.sciencedirect.com/science/article/pii/S0012365X97001039 But could not understand ...
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0 answers
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How to solve this problem using the Maximum matching algorithm for general graph?

For any general graph G(V,E) , the maximum matching can be calculated in O(√V.|E|) time using the following algorithm : https://www.researchgate.net/publication/...
0 votes
1 answer
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Perfect Matching in Bipartite Graph with mutually exclusive edges

Problem I would to solve Perfect Matching in Bipartite Graph Problem where some edges are mutually exclusive. Example Left vertices: $a,b,c$ Right vertices: $x,y,z$ Edges: $(a,x),(a,y),(b,z),(c,y)...
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4 votes
1 answer
312 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 $V'$?
2 votes
1 answer
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Hardness of a scheduling/assignment problem

I am trying to prove the hardness of the following problem. This problem is from google hashcode, qualification-round, 2020. Hier is a brief description of the problem. Given a list or libraries and ...
7 votes
1 answer
247 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 ...
3 votes
1 answer
167 views

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 $k \leq n$. ...
1 vote
1 answer
642 views

How to find maximum matching edges in undirected tree

Let $B$ be an undirected tree with $|V|$ nodes given as adjacency list. I want to develop a greedy algorithm using pseudo code to find a maximal matching in runtime $\mathcal{O}(|V|)$. My approach: ...
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3 votes
1 answer
56 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 ...
5 votes
1 answer
364 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 ...
2 votes
2 answers
500 views

Bipartite graph minimal amount of vertices required

I have a bipartite graph made of two sets (SET 1 and SET 2) and I want to determine how many vertices from the ...
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3 votes
0 answers
248 views

What is the exact time complexity of randomized Kuhn's algorithm?

Please, read the whole question before answering, the exact details of the implementation are important. Suppose that you want to find largest cardinality bipartite matching in bipartite graph with $...
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1 vote
1 answer
149 views

Maximum-cardinality matching in unbalanced bipartite graphs

Let $G = (X+Y, E)$ be a bipartite graph, and suppose we want to find a maximum-cardinality matching in $G$. The Hopcroft-Karp algorithm runs in time $O(|E|\sqrt{|V|})$, where here $|V| = |X|+|Y|$. So ...
1 vote
1 answer
70 views

Student Course Allocation Problem with Many Constraints [closed]

Problem statement In an university, there are $t$ course categories, $m$ courses, $n$ sections, $p$ students. $i$-th section has: A student capacity: $cap_i$. Two lecture timings. (Formally, each ...
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