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Planar Graph Perfect Matching is in NC [article]

Nima Anari, Vijay V. Vazirani
2018 arXiv   pre-print
Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it?  ...  In this paper, we give an NC algorithm for finding a perfect matching in a planar graph. Our algorithm uses the above-stated fact about counting matchings in a crucial way.  ...  They gave an NC algorithm for finding a perfect matching in bipartite planar graphs using a flow-based approach.  ... 
arXiv:1709.07822v4 fatcat:acnxoaaulnf7la3onera326kvu

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

Samir Datta, Raghav Kulkarni, Sambuddha Roy
2009 Theory of Computing Systems  
It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in non-bipartite planar graphs, which has been open for a long time.  ...  We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique.  ...  Matchings in Planar Graphs and Deterministic Isolation The situation for planar graphs is interesting because of the fact that counting the number of perfect matchings in planar graph is in NC ([K67, V88  ... 
doi:10.1007/s00224-009-9204-8 fatcat:2pgywowsxfe2deajsecwfgfhie

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs [article]

Samir Datta, Raghav Kulkarni, Sambuddha Roy
2008 arXiv   pre-print
It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in non-bipartite planar graphs, which has been open for a long time.  ...  We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique.  ...  Matchings in Planar Graphs and Deterministic Isolation The situation for planar graphs is interesting because of the fact that counting the number of perfect matchings in planar graph is in NC ([K67, V88  ... 
arXiv:0802.2850v1 fatcat:pktp2wlwive6xpqva5oa6ifng4

NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs [article]

David Eppstein, Vijay V. Vazirani
2020 arXiv   pre-print
Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we obtain NC algorithms for perfect matching in any minor-closed graph family  ...  In 1988, Vazirani gave an NC algorithm for computing the number of perfect matchings in K_3,3-minor-free graphs by building on Kasteleyn's scheme for planar graphs, and stated that this "opens up the possibility  ...  Acknowledgements The research of David Eppstein was supported in part by NSF grants CCF-1618301 and CCF-1616248. The research of Vijay Vazirani was supported in part by NSF grant CCF-1815901.  ... 
arXiv:1802.00084v2 fatcat:zlzad5kkynhpvjyuhjjluroh7a

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.  ...  For bipartite graphs it is in NC [19, 14] , and for planar graphs it is also in NC [30] . It is an open problem whether the unique perfect matching problem is in NC.  ... 
doi:10.1007/978-3-642-32589-2_43 fatcat:b4l4n4ljenh4fo6c3phwpekiia

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.  ...  For bipartite graphs it is in NC [19, 14] , and for planar graphs it is also in NC [30] . It is an open problem whether the unique perfect matching problem is in NC.  ... 
doi:10.1145/2934310 fatcat:5wyua5c4bjdxhg5yfnmfbwf54u

Perfect Matching in General vs. Cubic Graphs: A Note on the Planar and Bipartite Cases

E. Bampis, A. Giannakos, A. Karzanov, Y. Manoussakis, I. Milis
2000 RAIRO - Theoretical Informatics and Applications  
Afrati for the helpful comments on this work, and the anonymous référées for their precious remarks and suggestions that led to essential changes in the original manuscript.  ...  be in NC for planar graphs [17] (this last result implies an NC algorithm for finding a perfect matching in planar-bipartite graphs).  ...  and planar cases, would imply directly an NC algorithm for finding a perfect matching problem in any bipartite graph.  ... 
doi:10.1051/ita:2000108 fatcat:zoiyrb5o75fd7ouadeqql5s6ki

NC algorithms for computing the number of perfect matchings in K3,3-free graphs and related problems

Vijay V. Vazirani
1989 Information and Computation  
We show that the problem of computing the number of perfect matchings in K3.3free graphs is in NC.  ...  In this paper, we give a parallel algorithm for orienting K",-free graphs, thereby showing that the problem of computing the number of perfect matchings in K",-free graphs is in NC.  ...  ACKNOWLEDGMENTS First, thanks to Dave Johnson for accidentally starting me off by informing me, in a different context, of Hall's and Asano's results.  ... 
doi:10.1016/0890-5401(89)90017-5 fatcat:bfbye4qwrncztlc2s5cc22tv4a

NC Algorithms for Weighted Planar Perfect Matching and Related Problems [article]

Piotr Sankowski
2018 arXiv   pre-print
The main results of this paper are NC algorithms for the following problems: - minimum weight perfect matching in G, - maximum cardinality and maximum weight matching in G when G is bipartite, - maximum  ...  Consider a planar graph G=(V,E) with polynomially bounded edge weight function w:E→ [0, poly(n)].  ...  By Lemma 11 and 13 the recursion depth in Algorithm 2 is O(log 2 n) thus: Corollary 14. A perfect matching in a planar graph can be computed in NC.  ... 
arXiv:1709.07869v4 fatcat:3fs2gr2xjraspnllcnxhinr2im

Improved Bounds for Bipartite Matching on Surfaces

Samir Datta, Arjun Gopalan, Raghav Kulkarni, Raghunath Tewari, Marc Herbstritt
2012 Symposium on Theoretical Aspects of Computer Science  
We exhibit the following new upper bounds on the space complexity and the parallel complexity of the Bipartite Perfect Matching (BPM) problem for graphs of small genus: (1) BPM in planar graphs is in UL  ...  (improves upon the SPL bound from [7]); (2) BPM in constant genus graphs is in NL (orthogonal to the SPL bound from [8]); (3) BPM in poly-logarithmic genus graphs is in NC; (extends the NC bound for O  ...  Acknowledgement We would like to thank Prajakta Nimbhorkar for discussion in the initial stages of the work, in particular for pointing out that the Neg-Cycle problem is in NL.  ... 
doi:10.4230/lipics.stacs.2012.254 dblp:conf/stacs/DattaGKT12 fatcat:p2hinlmayfchffc4lvkmw77omu

Circuit Complexity of Bounded Planar Cutwidth Graph Matching [article]

Aayush Ojha, Raghunath Tewari
2018 arXiv   pre-print
Recently, perfect matching in bounded planar cutwidth bipartite graphs () was shown to be in ACC^0 by Hansen et al.. They also conjectured that the problem is in AC^0.  ...  Our results also imply a better lower bound for perfect matching in general bounded planar cutwidth graphs.  ...  He would also like to thank Jayalal Sarma for inviting him to the PhD defence of Balagopal Komarath where he gained more insight into the area in general.  ... 
arXiv:1801.00906v1 fatcat:gplm3zylmbdszidy4cg6ecqtfa

Seeking a Vertex of the Planar Matching Polytope in NC [chapter]

Raghav Kulkarni, Meena Mahajan
2004 Lecture Notes in Computer Science  
For planar bipartite graphs, finding a perfect matching when one exists can also be done in NC [8, 7] .  ...  For planar graphs, counting the number of perfect matchings (and hence determining whether there exists a perfect matching) can be done in NC [4, 10] .  ...  For planar graphs, the number of perfect matchings can be found in NC [4, 10] .  ... 
doi:10.1007/978-3-540-30140-0_43 fatcat:sa5egs4zxndqdbrpjspnvhtxkm

On the Matching Problem for Special Graph Classes

Thanh Minh Hoang
2010 2010 IEEE 25th Annual Conference on Computational Complexity  
Note that SPL, ⊕L, C = L, and L C=L are contained in NC 2 . Moreover, we show that the problem of computing a maximum matching for bipartite planar graphs is in L C=L .  ...  This solves Open Question 4.7 stated in the STACS'08-paper by Datta, Kulkarni, and Roy [DKR08] where it is asked whether computing a maximum matching even for bipartite planar graphs can be done in NC.  ...  Computing a perfect matching for bipartite planar graphs is known to be in SPL [DKR08] . Since SPL ⊆ C = L ⊆ L C=L , the maximum matching problem for bipartite planar graphs is in L C=L .  ... 
doi:10.1109/ccc.2010.21 dblp:conf/coco/Hoang10 fatcat:kk25ftl6brgdpm3o4be3sn5sxy

Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm [article]

Andreas Björklund, Thore Husfeldt
2018 arXiv   pre-print
Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a generalisation of S2DP, and fast algorithms for counting perfect matchings in planar graphs.  ...  In contrast, the randomized polynomial time algorithm by Björklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions.  ...  This work was supported in part by the Swedish Research Council grant VR-2016-03855, "Algebraic Graph Algorithms".  ... 
arXiv:1806.07586v1 fatcat:2xezuddxmbfdvljjxnb5ogujmi

Circuit Complexity of Properties of Graphs with Constant Planar Cutwidth [chapter]

Kristoffer Arnsfelt Hansen, Balagopal Komarath, Jayalal Sarma, Sven Skyum, Navid Talebanfard
2014 Lecture Notes in Computer Science  
bounded planar cutwidth graphs.  ...  We study the complexity of several of the classical graph decision problems in the setting of bounded cutwidth and how imposing planarity affects the complexity.  ...  The complexity of deciding if a graph has a perfect matching is still not known. It belongs to P, but it is an open problem if it belongs to NC.  ... 
doi:10.1007/978-3-662-44465-8_29 fatcat:zn26udh7i5e63m6e7t3ltfzoou
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