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Perfect Matchings via Uniform Sampling in Regular Bipartite Graphs [article]

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2008 arXiv   pre-print
In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph.  ...  We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a d-regular bipartite graph independently with a probability p = O(n n/d^2) then the resulting graph has a perfect  ...  Acknowledgments We thank Rajat Bhattacharjee for many helpful discussions in the early stages of this work.  ... 
arXiv:0811.2457v1 fatcat:varj5nmfprezxklic75ggteauy

Perfect Matchings via Uniform Sampling in Regular Bipartite Graphs [chapter]

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2009 Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms  
In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph.  ...  We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a d-regular bipartite graph independently with a probability p = O( n ln n d 2 ) then the resulting graph has  ...  Acknowledgments We thank Rajat Bhattacharjee for many helpful discussions in the early stages of this work.  ... 
doi:10.1137/1.9781611973068.2 fatcat:6d7xpfilpnbircm2dabtjwla74

Perfect matchings via uniform sampling in regular bipartite graphs

Ashish Goel, Michael Kapralov, Sanjeev Khanna
2010 ACM Transactions on Algorithms  
In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph.  ...  We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a d-regular bipartite graph independently with a probability p = O( n ln n d 2 ) then the resulting graph has  ...  Acknowledgments We thank Rajat Bhattacharjee for many helpful discussions in the early stages of this work.  ... 
doi:10.1145/1721837.1721843 fatcat:7gotcx4ucjblndgg7f5lapvnja

Approximating the permanent of graphs with large factors

Paul Dagum, Michael Luby
1992 Theoretical Computer Science  
We characterize the complexity of counting the number of perfect matchings in classes of graphs parameterized by factor size.  ...  The factor size of G, A is the maximum number of edge disjoint perfect matchings in G.  ...  Fig. 3 . 3 Theorem 6.2. (1) Exact counting of perfect matchings in 3-regular bipartite graphs is#P-complete.  ... 
doi:10.1016/0304-3975(92)90234-7 fatcat:o4rpq26chbhd3e4zbrrgo2om3m

Perfect Matchings in Õ(n^1.5) Time in Regular Bipartite Graphs [article]

Ashish Goel and Michael Kapralov and Sanjeev Khanna
2009 arXiv   pre-print
To obtain this result, we design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in a regular bipartite graph to the same problem on a subsampled bipartite  ...  We consider the well-studied problem of finding a perfect matching in d-regular bipartite graphs with 2n vertices and m = nd edges.  ...  Acknowledgments: We would like to thank Rajat Bhattacharjee for many useful discussions on precursors to this work, and Michel Goemans for suggesting an alternate proof mentioned in remark 3.9.  ... 
arXiv:0902.1617v2 fatcat:6fci7uvdrjgblbjdga63yrwgpy

Marathon: An Open Source Software Library for the Analysis of Markov-Chain Monte Carlo Algorithms

Steffen Rechner, Annabell Berger, Hans A Kestler
2016 PLoS ONE  
We use marathon to investigate the quality of several bounding methods on four well-known Markov chains for sampling perfect matchings and bipartite graph realizations.  ...  In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for random sampling of combinatorial objects.  ...  The problem of uniformly sampling a perfect matching in a bipartite graph and of uniformly sampling a bipartite graph realization for given vertex degrees.  ... 
doi:10.1371/journal.pone.0147935 pmid:26824442 pmcid:PMC4732767 fatcat:3yh6dweanbckbehqldp5xtvipq

Exact Sampling from Perfect Matchings of Dense Regular Bipartite Graphs

Mark Huber
2005 Algorithmica  
The problem of counting the number of perfect matchings in these types of graphs is P complete.  ...  We present the first algorithm for generating random variates exactly uniformly from the set of perfect matchings of a bipartite graph with a polynomial expected running time over a nontrivial set of graphs  ...  Given a bipartite graph (V, E) with n nodes in each of the two partitions (so |V | = 2n), a perfect matching is a subset of edges such that each node is incident to exactly one edge.  ... 
doi:10.1007/s00453-005-1175-9 fatcat:n5ynlfcahvchtd3dypuxrjwk4i

Broder's Chain Is Not Rapidly Mixing [article]

Annabell Berger, Steffen Rechner
2014 arXiv   pre-print
We prove that Broder's Markov chain for approximate sampling near-perfect and perfect matchings is not rapidly mixing for Hamiltonian, regular, threshold and planar bipartite graphs, filling a gap in the  ...  graph is known.  ...  Note that exact counting and uniform sampling perfect matchings in planar graphs is efficiently possible [Kas61, Edm65] .  ... 
arXiv:1404.4249v1 fatcat:t52tnnbmirha3oq3s2z2h6tgxe

On realizations of a joint degree matrix

Éva Czabarka, Aaron Dutle, Péter L. Erdős, István Miklós
2015 Discrete Applied Mathematics  
This was claimed before, but there is an error in the previous proof, which we illustrate by example.  ...  Finally, we address some of the issues concerning the mixing time of the corresponding MCMC method to sample uniformly from these realizations.  ...  Take a perfect matching on each bipartite graph (this is called a configuration).  ... 
doi:10.1016/j.dam.2014.10.012 fatcat:md4fzuchmnbe5g4iotptiy6mou

A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries

Mark Jerrum, Alistair Sinclair, Eric Vigoda
2004 Journal of the ACM  
The Sampling Algorithm As explained in the previous section, our goal now is to design an efficient (almost) uniform sampling algorithm for perfect matchings in a bipartite graph G = G A .  ...  Now, Broder [1986] has demonstrated how an almost uniform sampler for perfect matchings in a bipartite graph may be converted into an FPRAS.  ... 
doi:10.1145/1008731.1008738 fatcat:wnm4wuvuqve3xflqh3prmh5p7e

A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries

Mark Jerrum, Alistair Sinclair, Eric Vigoda
2001 Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01  
The Sampling Algorithm As explained in the previous section, our goal now is to design an efficient (almost) uniform sampling algorithm for perfect matchings in a bipartite graph G = G A .  ...  Now, Broder [1986] has demonstrated how an almost uniform sampler for perfect matchings in a bipartite graph may be converted into an FPRAS.  ... 
doi:10.1145/380752.380877 dblp:conf/stoc/JerrumSV01 fatcat:6gh23dco4vawraaitbbrcarz3i

An Experimental Study of Algorithms for Online Bipartite Matching [article]

Allan Borodin, Christodoulos Karavasilis, Denis Pankratov
2018 arXiv   pre-print
Greediness is by far the most important property of online algorithms for bipartite matching.  ...  We perform an experimental study of algorithms for online bipartite matching under the known i.i.d. input model with integral types.  ...  Consecutive pairs of blocks along the rope are connected alternately by perfect matchings and random bipartite graphs of average degree d − 1, beginning and ending with perfect matchings.  ... 
arXiv:1808.04863v1 fatcat:2cxtryl53fdvjjwcprhxswovdm

Bethe Learning of Conditional Random Fields via MAP Decoding [article]

Kui Tang, Nicholas Ruozzi, David Belanger, Tony Jebara
2015 arXiv   pre-print
Our algorithm outperforms existing methods in experiments involving image segmentation, matching problems from vision, and a new dataset of university roommate assignments.  ...  Many machine learning tasks can be formulated in terms of predicting structured outputs.  ...  We sample 10 × 10 bipartite matchings from the distribution (11).  ... 
arXiv:1503.01228v1 fatcat:3c2t2zgzqnhfvnizjnsdoft5je

The Perfect Matching Reconfiguration Problem

Marthe Bonamy, Nicolas Bousquet, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Arnaud Mary, Moritz Mühlenthaler, Kunihiro Wasa, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other?  ...  We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases.  ...  Sampling random perfect matchings The problem of sampling or enumerating perfect matchings in a graph received considerable attention (see, e.g., [31] ).  ... 
doi:10.4230/lipics.mfcs.2019.80 dblp:conf/mfcs/BonamyBHIKMMW19 fatcat:fa6svucqxvfqdju2qpywe7of6i

Counting vertices of integral polytopes defined by facets [article]

Heng Guo, Mark Jerrum
2022 arXiv   pre-print
The result follows from Valiant's classical result that counting perfect matchings in a bipartite graph is #P-complete [22] . 2 The perfect 2-matching polytope In this section we see that by going a little  ...  Thus, counting perfect matchings in a bipartite graph can be reduced to counting vertices of an easily computable and easily described polytope.  ...  Suppose u h = v h is the lowest common ancestor of u and v in T .  ... 
arXiv:2105.01469v2 fatcat:sxwbv3qiwff27mqnorcdfesehq
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