Results 1  10
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14
Accidental Algorithms
 In Proc. 47th Annual IEEE Symposium on Foundations of Computer Science 2006
, 2004
"... Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in onetoone, or possibly onetomany, fashion. In this paper we propose a new ..."
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Cited by 58 (2 self)
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Complexity theory is built fundamentally on the notion of efficient reduction among computational problems. Classical reductions involve gadgets that map solution fragments of one problem to solution fragments of another in onetoone, or possibly onetomany, fashion. In this paper we propose a new kind of reduction that allows for gadgets with manytomany correspondences, in which the individual correspondences among the solution fragments can no longer be identified. Their objective may be viewed as that of generating interference patterns among these solution fragments so as to conserve their sum. We show that such holographic reductions provide a method of translating a combinatorial problem to finite systems of polynomial equations with integer coefficients such that the number of solutions of the combinatorial problem can be counted in polynomial time if one of the systems has a solution over the complex numbers. We derive polynomial time algorithms in this way for a number of problems for which only exponential time algorithms were known before. General questions about complexity classes can also be formulated. If the method is applied to a #Pcomplete problem then polynomial systems can be obtained the solvability of which would imply P #P = NC2. 1
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 41 (16 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 20 (8 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
On Symmetric Signatures in Holographic Algorithms
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 135 (2006)
, 2006
"... 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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Cited by 14 (9 self)
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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.
Holographic algorithms: the power of dimensionality resolved
 In: Automata, Languages and Programming. In: Lecture Notes in Comput. Sci
, 2007
"... Valiant’s theory of holographic algorithms is a novel methodology to achieve exponential speedups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that ..."
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Cited by 11 (4 self)
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Valiant’s theory of holographic algorithms is a novel methodology to achieve exponential speedups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that 2dimensional bases are universal for holographic algorithms. 1
Some Observations on Holographic Algorithms
"... Abstract. We define the notion of diversity for families of finite functions, and express the limitations of a simple class of holographic algorithms in terms of limitations on diversity. We go on to describe polynomial time holographic algorithms for computing the parity of the following quantities ..."
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Cited by 9 (1 self)
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Abstract. We define the notion of diversity for families of finite functions, and express the limitations of a simple class of holographic algorithms in terms of limitations on diversity. We go on to describe polynomial time holographic algorithms for computing the parity of the following quantities for degree three planar undirected graphs: the number of 3colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case the parity can be computed for any slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes. These holographic algorithms use bases of three components, rather than two. 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 8 (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
Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
, 1007
"... We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a c ..."
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Cited by 7 (3 self)
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We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of Bürgisser on the VNPcompleteness of the partial permanent. In particular, we show that the partial permanent cannot be VNPcomplete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.
Holographic algorithms: . . .
"... Valiant’s theory of holographic algorithms is a novel methodology to achieve exponential speedups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that ..."
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Cited by 4 (2 self)
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Valiant’s theory of holographic algorithms is a novel methodology to achieve exponential speedups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that 2dimensional bases are universal for holographic algorithms.