# Questions tagged [matching]

A matching is a subset of the edges of a graph, such that no edge in the subset shares a vertex with another.

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### Alternative criterion for approximate maximum-weight perfect matching algorithms

I have asked this question in cs.stackeschange, and it was recommended to me that I asked it here. Is there any literature on approximate maximum-weight perfect matchings where the approximation ...
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### Counting matchings on 3-regular bipartite graphs

What I call a graph here allows parallel edges. Is the following problem #P-hard: INPUT: a 3-regular bipartite graph $G$ OUTPUT: the number of matchings of $G$. It is known that counting matchings ...
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### Counting sum of parities of cycle covers in cubic, planar, bipartite graphs

Let $G$ be a cubic (i.e. every degree exactly three), planar, bipartite graph. By Hall's theorem its edges can be partitioned into three perfect matchings. Take any such partition $M_0,M_1,M_2$ and ...
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### Algorithm for K-best NON perfect bipartite matchings

I was reading this great article: https://core.ac.uk/download/pdf/82129717.pdf It solves a generalization of the maximum sum assignment problem by finding the k best assignments and not only the best....
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### Optimally fair stable matching

There's a nice post by Gil Kalai which outlines the inherent bias in stable matching algorithms quantitatively. In the traditional loyd shapeley algorithm for $n$ men and $n$ women, given randomly ...
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### Min weight perfect matching with even number of red edges

Consider a weighted graph with some red edges. We are interested in finding a perfect matching, such that the number of red edges is even, and under the previous constraints, the weight is minimized. ...
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### For each edge, find a matching that containing it and has maximum weight

Given a weighted graph $G=(V,E)$. For each edge $e\in E$, we are interested in finding a maximum weight matching over all matchings that contains edge $e$. If $G$ is bipartite, then this can be done ...
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### Does the following 2-rounds distributed algorithm approximates a maximal matching well?

Let $G$ be an undirected graph. I'm looking for a two-rounds distributed algorithm that matches as many vertex pairs as possible. Consider the following protocol for vertex $v$. Use a fair coin to ...
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### Is perfect matching for bipartite graph with no cycles unique?

Given a balanced bipartite graph that satisfies Hall's theorem (is non singular) then it shown that it has at least one perfect matching. My question is if the balanced bipartite graph is also ...
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### Maximum matching M with the condition G[M] is 2K_2-free

Is there anything in the literature close to the following problem: Given a bipartite graph $G(V,E)$ with balanced bipartition $\{U,W\}$ , does there exist a perfect matching $M$ in $G$ such ...
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### Optimizing Maximum Weighted Matching (Edmonds Blossom)

Background: I've ported Edmonds Blossom Algorithm with Maximum Weighted Matching to Java: https://github.com/simlu/EdmondsBlossom/blob/master/src/Blossom.java The original Python implementation ...
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### Is Permanent $+$-reducible?

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$. Is ...
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### Almost regular subhypergraph of hypergraph with large minimal degree

I am interested in knowing whether the following conjecture is true or not: For every $d \geq 1$ there exist constants $\delta,M_0 > 0$ such that the following holds for all $M \geq M_0$. ...
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### What is the connection between moments of Gaussians and perfect matchings of graphs?

Today, I heard the following statement in a talk: The 4th moment of a $1$-dimensional Gaussian distribution with mean $0$ and variance $1$ is the same as the number of perfect matchings of a ...
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### Complexity of Uniform Generation of Perfect Matchings

Jerrum,Valiant and Vazirani on their paper "Random generation of combinatorial structures from a uniform" (http://www.cc.gatech.edu/~vazirani/AppCount.pdf) talk about seeing problems related to ...
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### Maximum weight matching and submodular functions

Given a bipartite graph $G = (U \cup V, E)$ with positive weights let $f: 2^U \rightarrow \mathbb{R}$ with $f(S)$ equal to the maximum weight matching in the graph $G[S\cup V]$. Is it true that $f$ ...
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### Is there an extension to the stable roommates problem with multiple roommates per room?

The stable roommates problem presents a set of N two-person rooms and 2N would-be roommates with preferences over each other, and asks for a stable allocation of roommates to rooms (and, really, to ...
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### Is it known whether counting $q$-dimensional $p$-matching is $\#W$-Hard?

The $q$-Dimensional $p$-Matching is defined as follows: Given disjoint universes $U_1,\ldots,U_q$, think of an element in $U_1\times\ldots\times U_q$ as a set that contains exactly one element from ...
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### Fast algorithm for weighted bipartite matching problem

I have a set of $n$ agents and a set of $n$ tasks, and I need to assign each agent to exactly one task such that a cost is minimised. Some agents are incompatible with some tasks. I have an ...
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### Another algorithm for stable matching?

The only algorithm I have seen to compute a stable matching is the one by Gale an Shapley. Is there any other algorithm to compute a stable matching?
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### Christofides algorithm for directed graph

Is it possible to implement the Christofides algorithm for an directed Graph? Suppose you have an undirected Graph, in which every vertex has an edges in both ways to every other in the graph (not to ...
Consider the Hungarian Search algorithm for max/min weighted bipartite matching. Let G be a bipartite graph weighted by $w:E\rightarrow\mathbb{R}$ and let $M$ be a matching in $G$, let $S$ be a subset ...