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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 19 (10 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
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 (6 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
Bases Collapse in Holographic Algorithms
 Electronic Colloquium on Computational Complexity Report
, 2007
"... Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) c ..."
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Cited by 7 (2 self)
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Holographic algorithms are a novel approach to design polynomial time computations using linear superpositions. Most holographic algorithms are designed with basis vectors of dimension 2. Recently Valiant showed that a basis of dimension 4 can be used to solve in P an interesting (restrictive SAT) counting problem mod 7. This problem without modulo 7 is #Pcomplete, and counting mod 2 is NPhard. We give a general collapse theorem for bases of dimension 4 to dimension 2 in the holographic algorithms framework. We also define an extension of holographic algorithms to allow more general support vectors. Finally we give a Basis Folding Theorem showing that in a natural setting the support vectors can be simulated by bases of dimension 2. 1
Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP
"... Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a un ..."
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Cited by 4 (1 self)
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Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #Phard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary realvalued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #Phard on general graphs but tractable on planar graphs, or (3) those which are #Phard even on planar graphs. The classification criteria
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Cited by 2 (2 self)
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.
Signature Theory in Holographic Algorithms
"... In the theory of holographic algorithms proposed by Valiant, computation is expressed and processed in terms of signatures. We substantially develop the signature theory in holographic algorithms. This theory is developed in terms of drealizability and dadmissibility. For the class of 2realizable ..."
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Cited by 1 (0 self)
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In the theory of holographic algorithms proposed by Valiant, computation is expressed and processed in terms of signatures. We substantially develop the signature theory in holographic algorithms. This theory is developed in terms of drealizability and dadmissibility. For the class of 2realizable signatures we prove a Birkhofftype theorem which determines this class. It gives a complete structural understanding of the relationship between 2realizability and 2admissibility. This is followed by characterization theorems for 1realizability and 1admissibility. Finally, using this theory of general (i.e., unsymmetric) It is generally conjectured that many combinatorial problems in the class NP or #P are not computable in polynomial time. The prevailing opinion is that these problems seem to require the accounting or processing of exponentially many potential solution fragments to the problem. However it is rather natural, and it should not cause any surprise, that the answer to such a problem can in general be
A THEORY FOR VALIANT’S MATCHCIRCUITS (EXTENDED ABSTRACT)
, 2008
"... The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every k, the nonsingular character matrices of kbit matchgates form a group, extending the recen ..."
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The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every k, the nonsingular character matrices of kbit matchgates form a group, extending the recent work of Cai and Choudhary [1] of the same result for the case of k = 2, and that the single and the twobit matchgates are universal for matchcircuits, answering a question of Valiant [4].
www.stacsconf.org A THEORY FOR VALIANT’S MATCHCIRCUITS (EXTENDED ABSTRACT)
, 2008
"... Abstract. The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every k, the nonsingular character matrices of kbit matchgates form a group, extending ..."
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Abstract. The computational function of a matchgate is represented by its character matrix. In this article, we show that all nonsingular character matrices are closed under matrix inverse operation, so that for every k, the nonsingular character matrices of kbit matchgates form a group, extending the recent work of Cai and Choudhary [1] of the same result for the case of k = 2, and that the single and the twobit matchgates are universal for matchcircuits, answering a question of Valiant [4]. 1.
The Complexity of Planar Boolean #CSP with Complex Weights ⋆
"... Abstract. We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This ..."
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Abstract. We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3, 3) to counting the number of Eulerian orientations over planar 4regular graphs to show the latter is #Phard. This strengthens a theorem by Huang and Lu to the planar setting. 1
Electronic Colloquium on Computational Complexity, Report No. 135 (2006) On Symmetric Signatures in Holographic Algorithms
"... The most intriguing aspect of the new theory of matchgate computations and holographic algorithms by Valiant [12] [14] is that its reach and ultimate capability are wide open. The methodology produces unexpected polynomial time algorithms solving problems which seem to require exponential time. To s ..."
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The most intriguing aspect of the new theory of matchgate computations and holographic algorithms by Valiant [12] [14] is that its reach and ultimate capability are wide open. The methodology produces unexpected polynomial time algorithms solving problems which seem to require exponential time. To sustain our belief in P = NP, we must begin to develop a theory which captures the limit of expressibility and power of this new methodology. In holographic algorithms, symmetric signatures have been particularly useful. We give a complete characterization of these symmetric signatures over all bases of size 1. These improve previous results [4] where only symmetric signatures over the Hadamard basis (special basis of size 1) were obtained. This in particular confirms a conjecture by Valiant [18]. We also give a complete characterization of Boolean symmetric signatures over bases of size 1. Finally, it is an open problem whether signatures over bases of higher dimensions are strictly more powerful. The recent result by Valiant [17] seems to suggest that bases of size 2 might be indeed more powerful than bases of size 1. This result is with regard to a restrictive counting version of #SAT called #PlRtwMon3CNF. It is known that the problem is #Phard, and its mod 2 version is ⊕Phard. Yet its mod 7 version is solvable in polynomial time by holographic algorithms. This was accomplished by a suitable symmetric signature over a basis of size 2 [17]. We show that the same unexpected holographic algorithm can be realized over a basis of size 1. Furthermore we prove that 7 is the only modulus for which such an “accidental algorithm ” exists. Subject: Computational and structural complexity. 1