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Perfectf-matchings andf-factors in hypergraphs—A combinatorial approach

Isabel Beckenbach, Robert Scheidweiler
2017 Discrete Mathematics  
Keywords: perfect f -matchings in hypergraphs, f -factors in hypergraphs, mengerian hypergraph, balanced hypergraph, perfect hypergraph, Hall's Theorem  ...  We prove characterizations of the existence of perfect f -matchings in uniform mengerian and perfect hypergraphs. Moreover, we investigate the f -factor problem in balanced hypergraphs.  ...  to the perfect f -matching problem in balanced hypergraphs we had f (v) ≤ 4 for all v ∈ V .  ... 
doi:10.1016/j.disc.2017.05.005 fatcat:5dff6egarbfqbmwwyd57h37i2i

The Tight Cut Decomposition of Matching Covered Uniformable Hypergraphs [article]

Isabel Beckenbach, Meike Hatzel, Sebastian Wiederrecht
2019 arXiv   pre-print
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms.  ...  Moreover, we show how the tight cut decomposition leads to a decomposition of the perfect matching polytope of uniformable hypergraphs and that the recognition problem for tight cuts in uniformable hypergraphs  ...  We will add a new characterisation of balanced hypergraphs in terms of their perfect matching polytope.  ... 
arXiv:1812.05461v3 fatcat:li2ksb3xsfbehik44jbqjvcv2y

Computing the Partition Function for Perfect Matchings in a Hypergraph

ALEXANDER BARVINOK, ALEX SAMORODNITSKY
2011 Combinatorics, probability & computing  
We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.  ...  When the weights w S are positive and within a constant factor, fixed in advance, of each other, we present a simple polynomial time algorithm to approximate the sum within a polynomial in m factor.  ...  Theorem 1.5 allows us to distinguish in polynomial time between hypergraphs that have sufficiently many perfect matchings and hypergraphs that do not have nearly perfect matchings.  ... 
doi:10.1017/s0963548311000435 fatcat:kgg6u57hprbircrdmd5y4roz6u

Computing the partition function for perfect matchings in a hypergraph [article]

Alexander Barvinok, Alex Samorodnitsky
2011 arXiv   pre-print
We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.  ...  When the weights w_S are positive and within a constant factor, fixed in advance, of each other, we present a simple polynomial time algorithm to approximate the sum within a polynomial in m factor.  ...  Theorem 1.6 allows us to distinguish in polynomial time between hypergraphs that have sufficiently many perfect matchings and hypergraphs that do not have nearly perfect matchings.  ... 
arXiv:1009.2397v3 fatcat:l54q6qgld5adpgwotodalvzrba

Matchings in 3-uniform hypergraphs

Daniela Kühn, Deryk Osthus, Andrew Treglown
2013 Journal of combinatorial theory. Series B (Print)  
Fact Let B be a balanced bipartite graph on 6 vertices. Then either B contains a perfect matching; B ∼ = B 023 , B 033 , B 113 or; e(B) ≤ 4.  ...  Fact Let B be a balanced bipartite graph on 6 vertices. Then either B contains a perfect matching; B ∼ = B 023 , B 033 , B 113 or; e(B) ≤ 4.  ... 
doi:10.1016/j.jctb.2012.11.005 fatcat:em4lylugxjbcnmxjyhkc3zroxu

Perfect matchings in r-partite r-graphs

Ron Aharoni, Agelos Georgakopoulos, Philipp Sprüssel
2009 European journal of combinatorics (Print)  
We prove that under this condition H must have a perfect matching. This answers a question of Kuhn and Osthus.  ...  Suppose that there exist two sides of H, each satisfying the following condition: the degree of each legal (r-1)-tuple contained in the complement of this side is strictly larger than n/2.  ...  Suppose, furthermore, that the degree of every pair in (V 1 × V 2 ) is 1 and the degree of every pair in (V 1 × V 3 ) is at most 1 . Then there exists in H a matching of size n.  ... 
doi:10.1016/j.ejc.2008.02.011 fatcat:wheqdrillbdybjoizyyq2qtbcq

The Multilinear Polytope for Acyclic Hypergraphs

Alberto Del Pia, Aida Khajavirad
2018 SIAM Journal on Optimization  
Utilizing an equivalent hypergraph representation, we study the facial structure of the Multilinear polytope in conjunction with the acyclicity degree of the underlying hypergraph.  ...  As the Multilinear polytope for γ-acyclic hypergraphs may contain exponentially many facets in general, we present a highly efficient polynomial algorithm to solve the separation problem.  ...  [14] proved that a minimum-weight perfect matching in balanced hypergraphs can be obtained in polynomial time via solving a linear optimization problem.  ... 
doi:10.1137/16m1095998 fatcat:6roknqn6jne3zlxw4yhddzltvy

Matchings in 3-uniform hypergraphs of large minimum vertex degree

Andrew Treglown, Daniela Kühn, Deryk Osthus
2011 Electronic Notes in Discrete Mathematics  
We determine the minimum vertex degree that ensures a perfect matching in a 3uniform hypergraph.  ...  If the minimum vertex degree of H is greater than n−1 2 − 2n/3 2 , then H contains a perfect matching. This bound is tight and answers a question of Hàn, Person and Schacht.  ...  Fact 2. 1 Fig. 1 . 11 Let B be a balanced bipartite graph on 6 vertices. • If e(B) ≥ 7 then B contains a perfect matching. • If e(B) = 6 then either B contains a perfect matching or B ∼ = B 033 . • If  ... 
doi:10.1016/j.endm.2011.10.036 fatcat:ckcmpb562vbmhjle4o6tx3vedu

Page 5399 of Mathematical Reviews Vol. , Issue 89J [page]

1989 Mathematical Reviews  
Let f(G) and g(G) be the number of perfect matchings of a factorable graph G and the number of maximum matchings of a nonfactorable graph G, respectively.  ...  Larry Basen§Spiler (Mobile, AL) 89j:05058 05C70 Zheng, Mao Lin (PRC-BJ) On the perfect matchings and the maximum matchings of n-connected graphs. (Chinese. English summary) Comm. Appl. Math.  ... 

Matchings and Hamilton cycles in hypergraphs

Daniela Kühn, Deryk Osthus
2005 Discrete Mathematics & Theoretical Computer Science  
International audience It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching.  ...  that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices.  ...  Matchings in uniform hypergraphs The so called 'marriage theorem' of Hall provides a necessary and sufficient condition for the existence of a perfect matching in a bipartite graph.  ... 
doi:10.46298/dmtcs.3457 fatcat:bkevf5myg5b3vmf7e63fg4qquu

Node-balancing by edge-increments [article]

Friedrich Eisenbrand, Shay Moran, Rom Pinchasi, Martin Skutella
2015 arXiv   pre-print
In this paper we study several variants of this problem for graphs and hypergraphs.  ...  On the combinatorial side we show connections with fundamental results from matching theory such as Hall's Theorem and Tutte's Theorem.  ...  This follows by a reduction from 3-dimensional matching [11] . Thus deciding whether a hypergraph has a perfect matching is an NPcomplete problem.  ... 
arXiv:1504.06919v2 fatcat:swizjw2kp5g7vlscqzryv3hx6y

Node-Balancing by Edge-Increments [chapter]

Friedrich Eisenbrand, Shay Moran, Rom Pinchasi, Martin Skutella
2015 Lecture Notes in Computer Science  
In this paper we study several variants of this problem for graphs and hypergraphs.  ...  On the combinatorial side we show connections with fundamental results from matching theory such as Hall's Theorem and Tutte's Theorem.  ...  This follows by a reduction from 3-dimensional matching [11] . Thus deciding whether a hypergraph has a perfect matching is an NPcomplete problem.  ... 
doi:10.1007/978-3-662-48350-3_38 fatcat:szfkvoq4kjappe6fsdgal7zeeq

An Approximation Result for Matchings in Partitioned Hypergraphs [chapter]

Isabel Beckenbach, Ralf Borndörfer
2016 Operations Research Proceedings  
We investigate the matching and perfect matching polytopes of hypergraphs having a special structure, which we call partitioned hypergraphs.  ...  We show that the integrality gap of the standard LP-relaxation is at most 2 √ d for partitioned hypergraphs with parts of size ≤ d.  ...  The hypergraph assignment problem can also be seen as a perfect matching problem in a hypergraph having the following special structure: Definition.  ... 
doi:10.1007/978-3-319-28697-6_5 dblp:conf/or/BeckenbachB14 fatcat:vnroapptfjgm7g6qu4vhoouf4i

Perfect fractional matchings in k-out hypergraphs [article]

Pat Devlin, Jeff Kahn
2017 arXiv   pre-print
In particular, we show that for each r there is a k=k(r) such that the k-out r-uniform hypergraph on n vertices has a perfect fractional matching with high probability (i.e., with probability tending to  ...  This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings.  ...  A perfect matching in a hypergraph is a collection of edges partitioning the vertex set.  ... 
arXiv:1703.03513v1 fatcat:v5lyfmn55fedjb3s46nw3xnjc4

Dynamic load-balancing with variable number of processors based on graph repartitioning

Clement Vuchener, Aurelien Esnard
2012 2012 19th International Conference on High Performance Computing  
In this paper, we present a new graph repartitioning algorithm which accepts to dynamically change the number of processors, assuming the load is already balanced.  ...  These approaches can be very inefficient, especially in terms of resource consumption.  ...  In this paper, we focus on perfect communication matrix, which results from two perfectly balanced partitions, P and P .  ... 
doi:10.1109/hipc.2012.6507501 dblp:conf/hipc/VuchenerE12 fatcat:x4fgmwaxbjhgnnktcj2k6kmldy
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