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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 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
On Valiantâ€™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.
Classification of simple 2(11,3,3) designs
"... We present an orderly algorithm for classifying triple systems. Subsequently, we show that there exist exactly 7,038,699,746 nonisomorphic simple 2(11, 3, 3) designs. The method is also used to confirm the previously accomplished classifications of 2(8,3,6), 2(12, 3, 2) and 2(19, 3, 1) designs. ..."
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We present an orderly algorithm for classifying triple systems. Subsequently, we show that there exist exactly 7,038,699,746 nonisomorphic simple 2(11, 3, 3) designs. The method is also used to confirm the previously accomplished classifications of 2(8,3,6), 2(12, 3, 2) and 2(19, 3, 1) designs.
On the Spectrum of Simple T(v, 3, 2) Trades
"... In this paper, we determine the spectrum (the set of all possible volumes) of simple T(2, 3, v) trades for any even foundation size v. ..."
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In this paper, we determine the spectrum (the set of all possible volumes) of simple T(2, 3, v) trades for any even foundation size v.
DOI 10.1007/s0045300993833 Signature Theory in Holographic Algorithms
"... Abstract 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 2r ..."
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Abstract 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) signatures we give additional counting problems solvable in polynomial time by holographic algorithms.
Repeated blocks in indecomposable twofold extended triple systems
"... An extended triple system (a twofold extended triple system) with no idempotent element (ETS, TETS respectively) is a pair (V, B) where V is a vset and B is a collection of unordered triples, called blocks, of type {x,y,z} or {x,x,y}, such that each pair (whether distinct or not) is contained in ex ..."
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An extended triple system (a twofold extended triple system) with no idempotent element (ETS, TETS respectively) is a pair (V, B) where V is a vset and B is a collection of unordered triples, called blocks, of type {x,y,z} or {x,x,y}, such that each pair (whether distinct or not) is contained in exactly one (respectively, exactly two) blocks. For example, in the block {x, x, y}, the occurrence of the pair {x, y} is counted once. It is wellknown that an ETS ( v) of order v (ETS ( v)) exists if and only if v = = 0 (mod 3), and it is trivial to see that a TETS of order v (TETS(v)) exists if and only if v = = 0 (mod 3). If a TETS (v) contains two blocks b l, b 2 that are identical as subsets of V, then bi = b2 is said to be a repeated block. We are interested in the following question: Given v = = 0 (mod 3) and a nonnegative integer k, does there exist a TETS(v) with exactly k repeated blocks? This question is related to the intersection problem for ETSs, solved by Lo Faro in 1995. The same question with the additional condition that the TETS be indecomposable (that is, cannot have its blocks partitioned into two ETS) is also of interest. The purpose of this paper is to completely settle these questions.