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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  
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.  ...  We further show that there is no planarizing gadget for the Hamiltonian cycle problem.  ...  Acknowledgment We thank an anonymous referee for his helpful comments.  ... 
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  
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.  ...  We further show that there is no planarizing gadget for the Hamiltonian cycle problem.  ...  Acknowledgment We thank an anonymous referee for his helpful comments.  ... 
doi:10.1007/978-3-642-32589-2_43 fatcat:b4l4n4ljenh4fo6c3phwpekiia

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.  ...  Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals.  ...  We thank Radu Curticapean for making us aware of the fast algorithms for determinants of matrices with a planar structure. We are also grateful for many valuable comments by anonymous referees.  ... 
arXiv:1806.07586v1 fatcat:2xezuddxmbfdvljjxnb5ogujmi

Counting the Number of Perfect Matchings in K5-Free Graphs

Simon Straub, Thomas Thierauf, Fabian Wagner
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
In the case of K 5 -free graphs, this technique will not work because some K 5 -free graphs do not have a Pfaffian orientation.  ...  The number of perfect matchings in the planar components can be computed efficiently. The major task to be accomplished here is to calculate the overall number of perfect matchings from them.  ...  The third case is that C is non-planar and = M 8 . This case we handle slightly different. Namely we do not place the gadgets right now in C.  ... 
doi:10.1109/ccc.2014.15 dblp:conf/coco/StraubTW14 fatcat:c4vyv6qdubefxgk3zbmwgxsznu

Counting the Number of Perfect Matchings in K 5-Free Graphs

Simon Straub, Thomas Thierauf, Fabian Wagner
2015 Theory of Computing Systems  
In the case of K 5 -free graphs, this technique will not work because some K 5 -free graphs do not have a Pfaffian orientation.  ...  The number of perfect matchings in the planar components can be computed efficiently. The major task to be accomplished here is to calculate the overall number of perfect matchings from them.  ...  The third case is that C is non-planar and = M 8 . This case we handle slightly different. Namely we do not place the gadgets right now in C.  ... 
doi:10.1007/s00224-015-9645-1 fatcat:mrjwsoss6bekjpe2glvozbyqxi

Counting Maximum Matchings in Planar Graphs Is Hard [article]

Istvan Miklos, Miklos Kresz
2020 arXiv   pre-print
This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time.  ...  We also prove that counting non-necessarily perfect matchings in planar graphs is already #P-complete if the problem is restricted to bipartite graphs.  ...  For each vertex in G 1 which is not in the crossing gadget, the weight is set to 1.  ... 
arXiv:2001.01493v1 fatcat:ljztlxwzhbeifpn23raxbuc5fm

Planarity, Determinants, Permanents, and (Unique) Matchings

Samir Datta, Raghav Kulkarni, Nutan Limaye, Meena Mahajan
2010 ACM Transactions on Computation Theory  
We also relate the complexity of grid graph reachability to that of testing existence/uniqueness of a perfect matching in a planar bipartite graph. ⋆ A preliminary version of this paper appeared in [16  ...  The permanent or the determinant modulo 2 is complete for ⊕L, and we show that parity of paths in a layered grid graph (which is bimodal planar) is also complete for this class.  ...  Acknowledgement The authors gratefully thank the referees for detailed comments that helped improve the presentation in the paper.  ... 
doi:10.1145/1714450.1714453 fatcat:b2o7r5wbqrh6nlrhztofm4rlpq

Space complexity: what makes planar graphs special?

Samir Datta, Raghav Kulkarni
2013 Bulletin of the European Association for Theoretical Computer Science  
The design of efficient algorithms for planar graphs, as a field of research, is over forty year old and continues to be an exciting area.  ...  There are several new efficient algorithms for a variety of graph optimization problems that exploit planarity and, in general, the structure of low genus graphs and graphs with excluded minors.  ...  Recently there have been some results on proving that certain type of planarizing gadgets do not exist (see for instance [20] ).  ... 
dblp:journals/eatcs/DattaK13 fatcat:errya5hbabeg5bh2qph5yobtiy

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

Andreas Björklund, Thore Husfeldt, Michael Wagner
2018 International Symposium on Algorithms and Computation  
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.  ...  Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals.  ...  Acknowledgements We thank Radu Curticapean for making us aware of the fast algorithms for determinants of matrices with a planar structure.  ... 
doi:10.4230/lipics.isaac.2018.19 dblp:conf/isaac/BjorklundH18 fatcat:e2zziditzbeovdzxx2tqzj24oi

Plane augmentation of plane graphs to meet parity constraints [article]

J.C. Catana, A. García, J. Tejel, J. Urrutia
2020 arXiv   pre-print
Analogous complexity results are obtained, when the augmentation must be done by a plane topological perfect matching between the vertices not meeting their parities.  ...  In addition, when it is required that the augmentation is made by a plane geometric perfect matching between the vertices not meeting their parities, we also prove that this augmentation problem is NP-complete  ...  We do this analysis only if the edge (v i , v k ) does not belong to G.  ... 
arXiv:2007.11863v1 fatcat:3k2wfbphdrgwhdtyxarbbwxgte

The permanent, graph gadgets and counting solutions for certain types of planar formulas [article]

Christian Schridde
2012 arXiv   pre-print
Whenever this bipartite double cover (BDC) is planar, the number of perfect matchings can be counted in polynomial time using Kasteleyn's algorithm Kas67.  ...  We show that the property of circular planarity plays a crucial role to find the correct gadgets for a counting problem. To circumvent the -barrier, we switch over to fields Z/pZ, for a prime p > 2.  ...  Acknowledgements Thanks goes to all my colleagues for fruitful discussion about the topic and to Peter Shor, who pointed out to me the existence of the Desnanot-Jacobi identity by answering one of my questions  ... 
arXiv:1205.4124v1 fatcat:jrfhbznsx5cydpqwoo6im5jaqm

Augmenting Geometric Graphs with Matchings [article]

Alexander Pilz, Jonathan Rollin, Lena Schlipf, André Schulz
2020 arXiv   pre-print
For any n-vertex polygon, with n > 3, we show that such a matching with less than n/7 edges is not maximal, that is, it can be extended by another compatible matching edge.  ...  We prove NP-completeness of deciding whether there is such a perfect matching.  ...  We thank Kevin Buchin, Michael Hoffmann, Wolfgang Mulzer and Nadja Seiferth for helpful discussions.  ... 
arXiv:2008.08413v1 fatcat:griddm36lzh6jcl66lb3qtpktm

k-L(2,1)-labelling for planar graphs is NP-complete for k≥4

Nicole Eggemann, Frédéric Havet, Steven D. Noble
2010 Discrete Applied Mathematics  
In this paper, we show that it remains NP-complete for any k >= 4 by reduction from Planar Cubic Two-Colourable Perfect Matching.  ...  Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete. In this paper we give a proof of this.  ...  In order to establish our main result we will reduce Planar Cubic Two-Colourable Perfect Matching to Planar k-L(2, 1)-Labelling for planar graphs for each k ≥ 4.  ... 
doi:10.1016/j.dam.2010.06.016 fatcat:qiju6ezftfgdriu4aicnwqaii4

Parameterizing the Permanent: Hardness for K_8-minor-free graphs [article]

Radu Curticapean, Mingji Xia
2021 arXiv   pre-print
In the 1960s, statistical physicists discovered a fascinating algorithm for counting perfect matchings in planar graphs. Valiant later showed that the same problem is #P-hard for general graphs.  ...  This stirred up hopes that counting perfect matchings might be polynomial-time solvable for graph classes excluding any fixed minor H.  ...  In other words, every perfect matching M incurs a factor χ e,f (M ) := −1 {e, f } ⊆ M, 1 otherwise. (2) Note that a "perfectly planarizing" crossing gadget that does not introduce negative signs would  ... 
arXiv:2108.12879v1 fatcat:5xxn3mm42janjfcbc4kocfiswe

The Graphs of Stably Matchable Pairs [article]

David Eppstein
2020 arXiv   pre-print
We study the graphs formed from instances of the stable matching problem by connecting pairs of elements with an edge when there exists a stable matching in which they are matched.  ...  We also provide characterizations of graphs of stably matchable pairs that belong to certain classes of graphs, and of the lattices of stable matchings that can have graphs in these classes.  ...  This might not be true for other gadgets; for instance, it is not true of the clause gadgets.  ... 
arXiv:2010.09230v1 fatcat:i3epotc4wnay5jmybfg4wdhjdu
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