Valiant’s Holant Theorem and Matchgate Tensors

Cai, Jin-Yi; Choudhary, Vinay

doi:10.1007/11750321_24

Cited by 25 publications

(14 citation statements)

References 14 publications

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“…The proof of this theorem follows from general principles of contravariant and covariant tensors and their contractions [2]. In particular, if there is a holographic reduction from #H : G|R to #H : G |R , and there is a P-time algorithm for #H : G |R , then there is a P-time algorithm for #H : G|R.…”

Section: Theorem 21 (Valiant's Holant Theorem) Suppose In a Hologramentioning

confidence: 97%

Holographic algorithms by Fibonacci gates

Cai

Xia

2013

Linear Algebra and its Applications

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We introduce Fibonacci gates as a polynomial time computable primitive, and develop a theory of holographic algorithms based on these gates. The Fibonacci gates play the role of matchgates in Valiant's theory (Valiant (2008) [19]). They give rise to polynomial time computable counting problems on general graphs, while matchgates mainly work over planar graphs only. We develop a signature theory and characterize all realizable signatures for Fibonacci gates. For bases of arbitrary dimensions we prove a basis collapse theorem. We apply this theory to give new polynomial time algorithms for certain counting problems. We also use this framework to prove that some slight variations of these counting problems are #P-hard. Holographic algorithms with Fibonacci gates prove to be useful as a general tool for classification results of counting problems (dichotomy theorems [7])).

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Section: Theorem 21 (Valiant's Holant Theorem) Suppose In a Hologramentioning

confidence: 97%

Holographic algorithms by Fibonacci gates

Cai

Xia

2013

Linear Algebra and its Applications

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“…Such algorithms have been studied in depth and further developed by Cai et al [1][2][3][4][5][6][7][8].…”

Section: Historymentioning

confidence: 99%

Holographic algorithms without matchgates

Landsberg

Morton

Norine

2013

Linear Algebra and its Applications

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The theory of holographic algorithms, which are polynomial time algorithms for certain combinatorial counting problems, surprised the complexity community by showing certain problems, very similar to #P complete problems, were in fact in the class P. In particular, the theory produces algebraic tests for a problem to be in P. In this article we describe the geometric basis of these algorithms by (i) replacing the construction of graph fragments in the procedure by the direct construction of a skew symmetric matrix, and (ii) replacing the computation of weighted perfect matchings of an auxiliary graph by computing the Pfaffian of the directly constructed skew-symmetric matrix. This procedure indicates a more geometric approach to complexity classes. It also leads to more general constructions where one replaces the "Grassmann-Plücker identities" which test for admissibility by other algebraic tests. Natural problems treatable by these methods have been previously considered in a different context, and we present one such example.

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“…Following the framework in [4], a generator is a contravariant tensor of the form G = (g i 1 i 2 ...i n ) where the index i 1 i 2 . .…”

Section: Beyond Symmetric Signaturesmentioning

confidence: 99%

Holographic algorithms: From art to science

Cai

2011

Journal of Computer and System Sciences

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115

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We develop the theory of holographic algorithms initiated by Leslie Valiant. First we define a basis manifold. Then we characterize algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and give a polynomial time decision algorithm for the simultaneous realizability problem. These results enable one to decide whether suitable signatures for a holographic algorithm are realizable, and if so, to find a suitable linear basis to realize these signatures by an efficient algorithm. 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 #P-complete without the moduli. Going beyond symmetric signatures, we define d-admissibility and d-realizability for general signatures, and give a characterization of 2-admissibility and some general constructions of admissible and realizable families.

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Valiant’s Holant Theorem and Matchgate Tensors

Abstract: We propose matchgate tensors as a natural and proper language to develop Valiant's new theory of Holographic Algorithms. We give a treatment of the central theorem in this theory -the Holant Theorem -in terms of matchgate tensors. Some generalizations are presented.

Cited by 25 publications

References 14 publications

Holographic algorithms by Fibonacci gates

Holographic algorithms by Fibonacci gates

Holographic algorithms without matchgates

Holographic algorithms: From art to science

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