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32
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
Graph Homomorphisms with Complex Values: A Dichotomy Theorem
"... Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined as ZA(G) = Aξ(u),ξ(v), ξ:V →[m] (u,v)∈E where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting ..."
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Cited by 31 (14 self)
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Graph homomorphism problem has been studied intensively. Given an m × m symmetric matrix A, the graph homomorphism function is defined as ZA(G) = Aξ(u),ξ(v), ξ:V →[m] (u,v)∈E where G = (V, E) is any undirected graph. The function ZA(G) can encode many interesting graph properties, including counting vertex covers and kcolorings. We study the computational complexity of ZA(G) for arbitrary complex valued symmetric matrices A. Building on work by Dyer and Greenhill [6], Bulatov and Grohe [2], and especially the recent beautiful work by Goldberg,
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
Normal factor graphs and holographic transformations
 IEEE Trans. Inf. Theory
, 2011
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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.
Dichotomy for Holant Problems of Boolean Domain
"... Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ (F), where F is a set of constraint functions on Boolean variables and taking comp ..."
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Cited by 13 (4 self)
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Holant problems are a general framework to study counting problems. Both counting Constraint Satisfaction Problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ (F), where F is a set of constraint functions on Boolean variables and taking complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #Phard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems over the complex domain, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (nonzero) vector whose inner product with itself is zero.
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
From Holant To #CSP And Back: Dichotomy For Holant c Problems
"... We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted Hcolorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results fro ..."
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Cited by 10 (6 self)
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We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted Hcolorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results from one framework can be used to derive results in another, and this happens in both directions. Holographic reductions discover an underlying unity, which is only revealed when these counting problems are investigated in the complex domain C. We prove three complexity dichotomy theorems, leading to a general theorem for Holant c problems. This is the natural class of Holant problems where one can assign constants 0 or 1. More specifically, given any signature grid on G = (V, E) over a set F of symmetric functions, we completely classify the complexity to be in P or #Phard, according to F, of X Y fv(σ E(v)), σ:E→{0,1} v∈V where fv ∈ F ∪ {0, 1} (0, 1 are the unary constant 0, 1 functions). Not only is holographic reduction the main tool, but also the final dichotomy can be only naturally stated in the language of holographic transformations. The proof goes through another dichotomy theorem on boolean complex weighted