Open AccessJournal Article
Some Results on Matchgates and Holographic Algorithms.
Jin-Yi Cai,Vinay Choudhary +1 more
TLDR
In this article, a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit and his signature theory of planar-match gate/matchgrid was established, which unified the two theories in expressibility.Abstract:
We establish a 1-1 correspondence between Valiant's character theory of matchgate/matchcircuit [14] and his signature theory of planar-matchgate/matchgrid [16], thus unifying the two theories in expressibility. In [3], we had established a complete characterization of general matchgates, in terms of a set of useful Grassmann-Plucker identities. With this correspondence, we give a corresponding set of identities which completely characterizes planar-matchgates and their signatures. Applying this characterization we prove some negative results for holographic algorithms. On the positive side, we also give a polynomial time algorithm for a simultaneous node-edge deletion problem, using holographic algorithms. Finally we give characterizations of symmetric signatures realizable in the Hadamard basis.read more
Citations
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Proceedings ArticleDOI
Accidental Algorthims
TL;DR: It is shown that for the NP-complete general 3CNF problem no such elementary matchgrid algorithm can exist, however, that it remains open for many natural #Pcomplete problems whether such Elementary matchgrid algorithms exist, and for the general CNF problem whether non-elementary match grid algorithms exist.
Proceedings Article
Holographic algorithms
TL;DR: In this paper, the authors introduce a new notion of efficient reduction among computational problems, which allows for gadgets with many-to-many correspondences, and provide a method of translating a combinatorial problem to a family of finite systems of polynomial equations with integer coefficients.
Proceedings ArticleDOI
Holographic algorithms: from art to science
Jin-Yi Cai,Pinyan Lu +1 more
TL;DR: Going beyond symmetric signatures, d-admissibility and d-realizability for general signatures are defined, and the general machinery is able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers.
Journal ArticleDOI
computational complexity of counting problems on 3-regular planar graphs
Mingji Xia,Peng Zhang,Wenbo Zhao +2 more
TL;DR: A sufficient condition is given which guarantees that the coefficients of a homogeneous polynomial can be uniquely determined by its values on a recurrence sequence and it is shown that #3-Regular Bipartite Planar Vertex Covers is #P-complete.
Journal ArticleDOI
Holographic algorithms: From art to science
Jin-Yi Cai,Pinyan Lu +1 more
TL;DR: The theory of holographic algorithms is developed, d-admissibility and d-realizability for general signatures are defined, and some general constructions of admissible and realizable families are given.
References
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Book
The Theory of Error-Correcting Codes
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
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The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice
TL;DR: In this paper, the number of ways in which a finite quadratic lattice can be fully covered with given numbers of "horizontal" and "vertical" dimers is rigorously calculated by a combinatorial method involving Pfaffians.
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Combinatorial Matrix Theory
Richard A. Brualdi,H. J. Ryser +1 more
TL;DR: In this paper, the existence theorems for combinatorially constrained matrices are given for instance matrices, digraphs, bigraphs and Latin squares, as well as some special graphs.
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Dimer problem in statistical mechanics-an exact result
TL;DR: In this article, an important model of a system (e.g., solution or gas) containing diatomic molecules is that of a lattice occupied by "rigid dimers" (i.e., rigid disks).
Book ChapterDOI
The Statistics of Dimers on a Lattice
TL;DR: In this paper, the number of ways in which a finite quadratic lattice can be fully covered with given numbers of "horizontal" and "vertical" dimers is rigorously calculated by a combinatorial method involving Pfaffians.