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74
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Alternating Sign Matrices and Domino Tilings
 Journal of Algebraic Combinatorics
, 1992
"... We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order n but also provides information about the orient ..."
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Cited by 47 (12 self)
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We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order n but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb. 1
On the Relative Complexity of Approximate Counting Problems
, 2000
"... Two natural classes of counting problems that are interreducible under approximationpreserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reducibili ..."
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Cited by 36 (12 self)
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Two natural classes of counting problems that are interreducible under approximationpreserving reductions are: (i) those that admit a particular kind of ecient approximation algorithm known as an \FPRAS," and (ii) those that are complete for #P with respect to approximationpreserving reducibility. We describe and investigate not only these two classes but also a third class, of intermediate complexity, that is not known to be identical to (i) or (ii). The third class can be characterised as the hardest problems in a logically dened subclass of #P. Research Report 370, Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK. This work was supported in part by the EPSRC Research Grant \Sharper Analysis of Randomised Algorithms: a Computational Approach" and by the ESPRIT Projects RANDAPX and ALCOMFT. y dyer@scs.leeds.ac.uk, School of Computer Studies, University of Leeds, Leeds LS2 9JT, United Kingdom. z leslie@dcs.warwick.ac.uk, http://www.dcs.warw...
On the Theory of Pfaffian Orientations. I. Perfect Matchings and Permanents.
 Electronic Journal of Combinatorics
, 1998
"... Kasteleyn stated that the generating function of the perfect matchings of a graph of genus g may be written as a linear combination of 4 g Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute the permanent of a square matrix. ..."
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Cited by 28 (5 self)
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Kasteleyn stated that the generating function of the perfect matchings of a graph of genus g may be written as a linear combination of 4 g Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute the permanent of a square matrix.
Statistical Mechanics, ThreeDimensionality and NPcompleteness I. Universality of Intractability for the Partition Function of the Ising Model Across NonPlanar Lattices (Extended Abstract)
"... This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working ..."
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Cited by 28 (1 self)
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This work provides an exact characterization, across crystal lattices, of the computational tractability frontier for the partition functions of several Ising models. Our results show that beyond planarity computing partition functions is NPcomplete. We provide rigorous solutions to several working conjectures in the statistical mechanics literature, such as the CrossedBonds conjecture, and the impossibility to compute effectively the partition functions for any threedimensional lattice Ising model � these conjectures apply to the Onsager algebraic method, the Fermion operators method, and the combinatorial method based on Pfaffians. The fundamental results of the area, including those of Onsager, Kac, Feynman, Fisher, Kasteleyn, Temperley, Green, Hurst and more recently Barahona: for every Planar crystal lattice the partition functions for the nite sublattices can be computed in polynomialtime, paired with the results of this paper: for every NonPlanar crystal lattice computing the parition functions for the finite sublattices is NPcomplete, provide an exact characterization for several of the most studied Ising models. Our results settle at once, for several models, (1) the 2D nonplanar vs. 2D planar, (2) the nextnearest neighbour
Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 22 (11 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
Approximating the Number of MonomerDimer Coverings of a Lattice
 Journal of Statistical Physics
, 1996
"... The paper studies the problem of counting the number of coverings of a ddimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn and Temper ..."
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Cited by 19 (2 self)
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The paper studies the problem of counting the number of coverings of a ddimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn and Temperley solves the problem exactly in two dimensions when the number of monomers is zero (the dimer covering problem), but is not applicable in higher dimensions or in the presence of monomers. This paper presents the first provably polynomial time approximation algorithms for computing the number of coverings with any specified number of monomers in ddimensional rectangular lattices with periodic boundaries, for any fixed dimension d , and in twodimensional lattices with fixed boundaries. The algorithms are based on Monte Carlo simulation of a suitable Markov chain, and, in contrast to most Monte Carlo algorithms in statistical physics, have rigorously derived performance guarantees that do n...
On the Theory of Matchgate Computations
 Submitted. Also available at Electronic Colloquium on Computational Complexity Report
, 2007
"... Valiant has proposed a new theory of algorithmic computation based on perfect matchings and the Pfaffian. We study the properties of matchgates—the basic building blocks in this new theory. We give a set of algebraic identities which completely characterize these objects in terms of the GrassmannPl ..."
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Cited by 14 (5 self)
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Valiant has proposed a new theory of algorithmic computation based on perfect matchings and the Pfaffian. We study the properties of matchgates—the basic building blocks in this new theory. We give a set of algebraic identities which completely characterize these objects in terms of the GrassmannPlücker identities. In the important case of 4 by 4 matchgate matrices, which was used in Valiant’s classical simulation of a fragment of quantum computations, we further realize a group action on the character matrix of a matchgate, and relate this information to its compound matrix. Then we use Jacobi’s theorem to prove that in this case the invertible matchgate matrices form a multiplicative group. These results are useful in establishing limitations on the ultimate capabilities of Valiant’s theory of matchgate computations and his closely related theory of Holographic Algorithms. 1
Valiant’s Holant Theorem and Matchgate Tensors (Extended Abstract
 In Proceedings of TAMC 2006: Lecture Notes in Computer Science
"... Abstract We propose matchgate tensors as a natural and proper language to develop Valiant's newtheory of Holographic Algorithms. We give a treatment of the central theorem in this theorythe Holant Theoremin terms of matchgate tensors. Some generalizations are presented. 1 Background In a remarka ..."
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Cited by 13 (7 self)
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Abstract We propose matchgate tensors as a natural and proper language to develop Valiant's newtheory of Holographic Algorithms. We give a treatment of the central theorem in this theorythe Holant Theoremin terms of matchgate tensors. Some generalizations are presented. 1 Background In a remarkable paper, Valiant [9] in 2004 has proposed a completely new theory of Holographic Algorithms or Holographic Reductions. In this framework, Valiant has developed a most novel methodology of designing polynomial time (indeed NC2) algorithms, a methodology by which one can design a custom made process capable of carrying out a seemingly exponential computation with exponentially many cancellations so that the computation can actually be done in polynomial time. The simplest analogy is perhaps with Strassen's matrix multiplication algorithm [5]. Here the algorithm computes some extraneous quantities in terms of the submatrices, which do not directly appear in the answer yet only to be canceled later, but the purpose of which is to speedup computation by introducing cancelations. In the several cases such clever algorithms had been found, they tend to work in a linear algebraic setting, in particular the computation of the determinant figures prominently [8, 2, 6]. Valiant's new theory manages to create a process of custom made cancelation which gives polynomial time algorithms for combinatorial problems which do not appear to be linear algebraic. In terms of its broader impact in complexity theory, one can view Valiant's new theory as another algorithmic design paradigm which pushes back the frontier of what is solvable by polynomial time. Admittedly, at this early stage, it is still premature to say what drastic consequence it might have on the landscape of the big questions of complexity theory, such as P vs. NP. But the new theory has already been used by Valiant to devise polynomial time algorithms for a number of problems for which no polynomial time algorithms were known before.