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Planarizing Gadgets for Perfect Matching Do Not Exist

Rohit Gurjar, Arpita Korwar, Jochen Messner, Simon Straub, Thomas Thierauf
2016 ACM Transactions on Computation Theory  
To reduce a 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.  ... 
doi:10.1145/2934310 fatcat:5wyua5c4bjdxhg5yfnmfbwf54u

Planarizing Gadgets for Perfect Matching Do Not Exist [chapter]

Rohit Gurjar, Arpita Korwar, Jochen Messner, Simon Straub, Thomas Thierauf
2012 Lecture Notes in Computer Science  
To reduce a 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.  ... 
doi:10.1007/978-3-642-32589-2_43 fatcat:b4l4n4ljenh4fo6c3phwpekiia

DNA Sequencing, Eulerian Graphs, and the Exact Perfect Matching Problem [chapter]

Jacek Błażewicz, Piotr Formanowicz, Marta Kasprzak, Petra Schuurman, Gerhard J. Woeginger
2002 Lecture Notes in Computer Science  
What we prove is that this problem is polynomial time equivalent to the 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.  ... 
doi:10.1007/3-540-36379-3_2 fatcat:n2lgqxb4ozh67b4k3t7zt4iuxu

Insight into Voting Problem Complexity Using Randomized Classes [article]

Zack Fitzsimmons, Edith Hemaspaandra
2022 arXiv   pre-print
We show that this problem is equivalent to 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.  ... 
arXiv:2204.12856v2 fatcat:hh5wpbkyxvbofhbhfjv725fgny

PyMatching: A Python package for decoding quantum codes with minimum-weight perfect matching [article]

Oscar Higgott
2021 arXiv   pre-print
This paper introduces PyMatching, a fast open-source Python package for decoding quantum error-correcting codes with the minimum-weight 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.  ... 
arXiv:2105.13082v2 fatcat:ooahaiczgrelhh4gkep7mh6o4i

Exact Matching and the Top-k Perfect Matching Problem [article]

Nicolas El Maalouly, Lasse Wulf
2022 arXiv   pre-print
The Top-k 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.  ... 
arXiv:2209.09661v1 fatcat:53w62eivzbavjjml25w2zojasi

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  ... 
doi:10.1016/s1570-8667(03)00075-3 fatcat:icg7if3uingibmjwy2mjud4rwe

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

Jacek Błażewicz, Piotr Formanowicz, Marta Kasprzak, Petra Schuurman, Gerhard J. Woeginger
2007 Discrete Optimization  
What we prove is that this problem is polynomial time equivalent to the 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.  ... 
doi:10.1016/j.disopt.2006.07.004 fatcat:ynfziuoprzaubawbmv5aqjuwta

Optimizing over a slice of the bipartite matching polytope

Matthias Leclerc
1988 Discrete Mathematics  
We discuss a special case of the 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.  ... 
doi:10.1016/0012-365x(88)90143-4 fatcat:v4c3injpkzeg7lmvgqgdsqbkba

Fractional matching preclusion for restricted hypercube-like graphs [article]

Huazhong Lü, Tingzeng Wu
2019 arXiv   pre-print
The 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.  ... 
arXiv:1905.04631v1 fatcat:cy6xn35y3vh2hbvqpofbzehflq

The Number of Perfect Matchings in Möbius Ladders and Prisms [article]

R.S.Lekshmi and Douglas B. West (Zhejiang Normal University, Jinhua, China, and University of Illinois, Urbana, IL)
2020 arXiv   pre-print
We give the 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.  ... 
arXiv:2003.09602v2 fatcat:kqop4xi3jvclhj77ocu72oap7e

A faster hafnian formula for complex matrices and its benchmarking on a supercomputer [article]

Andreas Björklund, Brajesh Gupt, Nicolás Quesada
2019 arXiv   pre-print
We introduce new and simple algorithms for the calculation of the number of 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.  ... 
arXiv:1805.12498v3 fatcat:2hhpif74gnak7no3msgliesaay

An algebraic Monte-Carlo algorithm for the partition adjacency matrix realization problem

Éva Czabarka, László Székely, Zoltán Toroczkai, Shanise Walker
2021 Algebraic Statistics  
Here we formulate common generalizations of this problem and the 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?  ... 
doi:10.2140/astat.2021.12.115 fatcat:54vquaqyefeiloy7tiogypmct4

Almost Exact Matchings [chapter]

Raphael Yuster
2007 Lecture Notes in Computer Science  
We show how to count the number of 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.  ... 
doi:10.1007/978-3-540-74208-1_21 fatcat:vk2j6el5rze6xe2jycoz75x7em

Almost Exact Matchings

Raphael Yuster
2011 Algorithmica  
We show how to count the number of 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.  ... 
doi:10.1007/s00453-011-9519-0 fatcat:klmvpfttqzdnzap23aqnvtsoce
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