Holographic algorithms with unsymmetric signatures

Holographic algorithms were introduced by Valiant as a new methodology to derive polynomial time algorithms. Here information and computation are represented by exponential sums using the so-called signatures. These signatures express superpositions of perfect matchings, and are used to achieve exponential sized cancellations, and thereby exponential speedups. Most holographic algorithms so far used symmetric signatures. In this paper we use unsymmetric signatures to give some new holographic algorithms. We also prove a characterization theorem for a class of realizable un-symmetric signatures, each of which may be used to design new holographic algorithms.

[1]  Leslie G. Valiant,et al.  Holographic Circuits , 2005, ICALP.

[2]  Leslie G. Valiant,et al.  Completeness for Parity Problems , 2005, COCOON.

[3]  A. S. Hedayat,et al.  On Theory and Applications of BIB Designs with Repeated Blocks. , 1977 .

[4]  Jin-Yi Cai,et al.  Some Results on Matchgates and Holographic Algorithms , 2007, Int. J. Softw. Informatics.

[5]  László Lovász,et al.  Kneser's Conjecture, Chromatic Number, and Homotopy , 1978, J. Comb. Theory A.

[6]  David Lichtenstein,et al.  Planar Formulae and Their Uses , 1982, SIAM J. Comput..

[7]  Leslie G. Valiant,et al.  Quantum Circuits That Can Be Simulated Classically in Polynomial Time , 2002, SIAM J. Comput..

[8]  Leslie G. Valiant,et al.  Holographic Algorithms (Extended Abstract) , 2004, FOCS.

[9]  Leslie G. Valiant.  Expressiveness of matchgates , 2002, Theor. Comput. Sci..

[10]  Nathan Linial,et al.  Incidence Matrices of Subsets—A Rank Formula , 1981 .

[11]  Jin-Yi Cai,et al.  On Symmetric Signatures in Holographic Algorithms , 2007, STACS.

[12]  A. Hedayat,et al.  Note: Correction to "On Theory and Application of BIB Designs with Repeated Blocks" , 1979 .

[13]  Emanuel Knill,et al.  Fermionic Linear Optics and Matchgates , 2001, ArXiv.

[14]  M. Fisher,et al.  Dimer problem in statistical mechanics-an exact result , 1961 .

[15]  Jin-Yi Cai,et al.  Holographic Algorithms: The Power of Dimensionality Resolved , 2007, ICALP.

[16]  M. Jerrum.  Two-dimensional monomer-dimer systems are computationally intractable , 1987 .

[17]  Jin-Yi Cai,et al.  Valiant's Holant Theorem and matchgate tensors , 2007, Theor. Comput. Sci..

[18]  Shuo-Yen Robert Li,et al.  On the Structure of t-Designs , 1980, SIAM J. Algebraic Discret. Methods.

[19]  Jin-Yi Cai,et al.  Holographic algorithms: from art to science , 2007, STOC '07.

[20]  Leslie G. Valiant,et al.  Accidental Algorithms , 2006, FOCS.

[21]  Kazuo Murota,et al.  Matrices and Matroids for Systems Analysis , 2000 .

[22]  Jin-Yi Cai,et al.  Bases Collapse in Holographic Algorithms , 2007, Computational Complexity Conference.

[23]  Jirí Matousek,et al.  A Combinatorial Proof of Kneser’s Conjecture* , 2004, Comb..

[24]  P. W. Kasteleyn.  The Statistics of Dimers on a Lattice , 1961 .

[25]  Jin-Yi Cai,et al.  On the Theory of Matchgate Computations , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).