Bases Collapse in Holographic Algorithms

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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Bibliographic Details
Published inTwenty-Second Annual IEEE Conference on Computational Complexity (CCC'07) pp. 292 - 304
Main Authors Jin-Yi Cai, Pinyan Lu
Format Conference Proceeding
LanguageEnglish
Published IEEE 01.06.2007
Subjects
Algorithm design and analysis
Computational complexity
Computational modeling
Dynamic programming
Holography
Linear programming
Polynomials
Quantum computing
Tensile stress
Vectors
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ISBN9780769527802
0769527809
ISSN1093-0159
DOI10.1109/CCC.2007.6

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Summary: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 #P-complete, and counting mod 2 is NP-hard. 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.
ISBN:9780769527802
0769527809
ISSN:1093-0159
DOI:10.1109/CCC.2007.6