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Planarizing Gadgets for Perfect Matching Do Not Exist

2016
*
ACM Transactions on Computation Theory
*

To reduce a

doi:10.1145/2934310
fatcat:5wyua5c4bjdxhg5yfnmfbwf54u
*graph*problem to its planar version, a standard technique is to replace crossings*in*a drawing of the input*graph*by planarizing gadgets. ... We show unconditionally that such a reduction is not possible for the*perfect**matching*problem and also extend this to some other problems related to*perfect**matching*. ... That is, Mod 2*perfect**matching**in*bipartite*graphs*can be computed*in*NC. Therefore, Mod 2*perfect**matching*is unlikely to be*complete*for ⊕P. ...##
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Planarizing Gadgets for Perfect Matching Do Not Exist
[chapter]

2012
*
Lecture Notes in Computer Science
*

To reduce a

doi:10.1007/978-3-642-32589-2_43
fatcat:b4l4n4ljenh4fo6c3phwpekiia
*graph*problem to its planar version, a standard technique is to replace crossings*in*a drawing of the input*graph*by planarizing gadgets. ... We show unconditionally that such a reduction is not possible for the*perfect**matching*problem and also extend this to some other problems related to*perfect**matching*. ... That is, Mod 2*perfect**matching**in*bipartite*graphs*can be computed*in*NC. Therefore, Mod 2*perfect**matching*is unlikely to be*complete*for ⊕P. ...##
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DNA Sequencing, Eulerian Graphs, and the Exact Perfect Matching Problem
[chapter]

2002
*
Lecture Notes in Computer Science
*

What we prove is that this problem is polynomial time equivalent to the

doi:10.1007/3-540-36379-3_2
fatcat:n2lgqxb4ozh67b4k3t7zt4iuxu
*exact**perfect**matching*problem*in*bipartite*graphs*, which is another infamous combinatorial optimization problem of unknown computational ... The problem is to decide whether there exists a word of length that contains every word*in*S at least once as a subword, and does not contain any other subword of length k. ... The first problem is the*Exact**Perfect**Matching*problem that asks whether a given edge weighted*graph*possesses a*perfect**matching*with weight exactly equal to a given bound. ...##
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Insight into Voting Problem Complexity Using Randomized Classes
[article]

2022
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arXiv
*
pre-print

We show that this problem is equivalent to

arXiv:2204.12856v2
fatcat:hh5wpbkyxvbofhbhfjv725fgny
*Exact**Perfect*Bipartite*Matching*, and so CCRV for First-Last can be determined*in*random polynomial time. ... On the other hand, showing that CCRV for First-Last is*in*P would also show that*Exact**Perfect*Bipartite*Matching*is*in*P, which would solve a well-studied 40-year-old open problem. ... Acknowledgements This work was supported*in*part by NSF-DUE-1819546. We thank Lane Hemaspaandra for helpful discussions and the anonymous reviewers for helpful comments and suggestions. ...##
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PyMatching: A Python package for decoding quantum codes with minimum-weight perfect matching
[article]

2021
*
arXiv
*
pre-print

This paper introduces PyMatching, a fast open-source Python package for decoding quantum error-correcting codes with the minimum-weight

arXiv:2105.13082v2
fatcat:ooahaiczgrelhh4gkep7mh6o4i
*perfect**matching*(MWPM) algorithm. ... The decoding performance of local*matching*is almost identical to that of the standard MWPM decoder*in*practice, while reducing the computational complexity approximately quadratically. ... We acknowledge the use of the UCL Myriad High Performance Computing Facility (Myriad@UCL), and associated support services,*in*the*completion*of this work. ...##
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Exact Matching and the Top-k Perfect Matching Problem
[article]

2022
*
arXiv
*
pre-print

The Top-k

arXiv:2209.09661v1
fatcat:53w62eivzbavjjml25w2zojasi
*Perfect**Matching*Problem is the problem of finding a*perfect**matching*which maximizes the total weight of the k heaviest edges contained*in*it. ... The aim of this note is to provide a reduction of the*Exact**Matching*problem to the Top-k*Perfect**Matching*Problem. ...*Exact**Matching*(EM) Input: A*graph*G, where every edge is colored blue or red, and an integer k. Task: Decide whether there exists a*perfect**matching*M*in*G with exactly k red edges. ...##
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Subject Index

2003
*
Journal of Discrete Algorithms
*

*graphs*, 281; Optimal gossiping

*in*paths and cycles, 461 Interval routing The compactness of adaptive routing tables, 237 Leader election

*Exact*communication costs for consensus and leader

*in*a ... distance Approximate string

*matching*on Ziv-Lempel compressed text, 313; Applying an edit distance to the

*matching*of tree ring sequences

*in*den- drochronology, 367

*Exact*algorithm An efficient fixed-parameter ...

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A polynomial time equivalence between DNA sequencing and the exact perfect matching problem

2007
*
Discrete Optimization
*

What we prove is that this problem is polynomial time equivalent to the

doi:10.1016/j.disopt.2006.07.004
fatcat:ynfziuoprzaubawbmv5aqjuwta
*exact**perfect**matching*problem*in*bipartite*graphs*, which is another infamous combinatorial optimization problem of unknown computational ... The problem is to decide whether there exists a word of length that contains every word*in*S at least once as a subword, and does not contain any other subword of length k. ... The first problem is the*Exact**Perfect**Matching*problem that asks whether a given edge weighted*graph*possesses a*perfect**matching*with weight exactly equal to a given bound. ...##
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Optimizing over a slice of the bipartite matching polytope

1988
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Discrete Mathematics
*

We discuss a special case of the

doi:10.1016/0012-365x(88)90143-4
fatcat:v4c3injpkzeg7lmvgqgdsqbkba
*Exact*solvable. A good algorithm will be given.*Perfect**Matching*Problem, which is polynomially ...*In*this paper we will consider another special instance of*Exact**Perfect**Matching*for nwkir a polynomial algorithm exists. ...*In*case (b) it is easy to reduce the problem to a minimum weight*perfect**matching*problem by assigning a cost function*perfect**matching*of the*graph*. ...##
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Fractional matching preclusion for restricted hypercube-like graphs
[article]

2019
*
arXiv
*
pre-print

The

arXiv:1905.04631v1
fatcat:cy6xn35y3vh2hbvqpofbzehflq
*matching*preclusion number of a*graph*is the minimum number of edges whose deletion results*in*the*graph*with neither*perfect**matchings*nor almost*perfect**matchings*. ...*In*this paper, we obtain fractional*matching*preclusion number and fractional strong*matching*preclusion numbers of restricted hypercube-like*graphs*, which extend some known results. ...*In*[12] , the authors obtained fractional*perfect*(strong)*matching*preclusion number the*complete**graph*, the Petersen*graph*and the twisted cube. ...##
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The Number of Perfect Matchings in Möbius Ladders and Prisms
[article]

2020
*
arXiv
*
pre-print

We give the

arXiv:2003.09602v2
fatcat:kqop4xi3jvclhj77ocu72oap7e
*exact*formula for the number of*perfect**matchings**in*two families of 3-regular*graphs*. ... The 1970s conjecture of Lovász and Plummer that the number of*perfect**matchings**in*any 3-regular*graph*is exponential*in*the number of vertices was proved*in*2011 by Esperet, Kardoš, King, Král', and Norine ... Our purpose*in*this note is to determine the*exact*number of*perfect**matchings**in*two families of*graphs*that have been of interest*in*other contexts. ...##
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A faster hafnian formula for complex matrices and its benchmarking on a supercomputer
[article]

2019
*
arXiv
*
pre-print

We introduce new and simple algorithms for the calculation of the number of

arXiv:1805.12498v3
fatcat:2hhpif74gnak7no3msgliesaay
*perfect**matchings*of complex weighted, undirected*graphs*with and without loops. ... Our compact formulas for the hafnian and loop hafnian of n × n complex matrices run*in*O(n^3 2^n/2) time, are embarrassingly parallelizable and, to the best of our knowledge, are the fastest*exact*algorithms ... This is simply the set of*perfect**matchings*of a*complete**graph*with loops. ...##
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An algebraic Monte-Carlo algorithm for the partition adjacency matrix realization problem

2021
*
Algebraic Statistics
*

Here we formulate common generalizations of this problem and the

doi:10.2140/astat.2021.12.115
fatcat:54vquaqyefeiloy7tiogypmct4
*exact**matching*problem, and solve them with an algebraic Monte-Carlo algorithm that runs*in*polynomial time if the number of partition classes ... The graphical realization of a given degree sequence and given partition adjacency matrix simultaneously, is a relevant problem*in*data driven modeling of networks. ... Let us recall:*Exact**matching*problem. Given a*graph*G, whose edges are colored red or green, is there a*perfect**matching*with exactly m red edges*in*the*matching*? ...##
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Almost Exact Matchings
[chapter]

2007
*
Lecture Notes in Computer Science
*

We show how to count the number of

doi:10.1007/978-3-540-74208-1_21
fatcat:vk2j6el5rze6xe2jycoz75x7em
*exact**perfect**matchings**in*K 3,3 -minor free*graphs*(these include all planar*graphs*as well as many others)*in*O(n 3.19 ) worst case time. ...*In*the*exact**matching*problem we are given a*graph*G, some of whose edges are colored red, and a positive integer k. ... Acknowledgment We thank the referees for providing constructive comments and help*in*improving the contents of this paper. ...##
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Almost Exact Matchings

2011
*
Algorithmica
*

We show how to count the number of

doi:10.1007/s00453-011-9519-0
fatcat:klmvpfttqzdnzap23aqnvtsoce
*exact**perfect**matchings**in*K 3,3 -minor free*graphs*(these include all planar*graphs*as well as many others)*in*O(n 3.19 ) worst case time. ...*In*the*exact**matching*problem we are given a*graph*G, some of whose edges are colored red, and a positive integer k. ... Acknowledgment We thank the referees for providing constructive comments and help*in*improving the contents of this paper. ...
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