Holographic algorithms by Fibonacci gates

Abstract 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 (Cai et al. (2009) [7])).

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

[2]  Jin-Yi Cai,et al.  Holographic Algorithms with Matchgates Capture Precisely Tractable Planar_#CSP , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

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

[4]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[5]  P. W. Kasteleyn.  The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice , 1961 .

[6]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[7]  Jin-Yi Cai,et al.  Holographic algorithms: The power of dimensionality resolved , 2009, Theor. Comput. Sci..

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

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

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

[11]  Jin-Yi Cai,et al.  Basis Collapse in Holographic Algorithms , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

[12]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[13]  Jin-Yi Cai,et al.  Valiant's Holant Theorem and Matchgate Tensors , 2006, TAMC.

[14]  P. Dienes,et al.  On tensor geometry , 1926 .

[15]  Nadia Creignou,et al.  Complexity of Generalized Satisfiability Counting Problems , 1996, Inf. Comput..

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

[17]  Jin-Yi Cai,et al.  Holant problems and counting CSP , 2009, STOC '09.

[18]  G. Sposito,et al.  Graph theory and theoretical physics , 1969 .

[19]  Jin-Yi Cai,et al.  Holographic Algorithms by Fibonacci Gates and Holographic Reductions for Hardness , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

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

[21]  Peng Zhang,et al.  Computational complexity of counting problems on 3-regular planar graphs , 2007, Theor. Comput. Sci..

[22]  Salil P. Vadhan,et al.  The Complexity of Counting in Sparse, Regular, and Planar Graphs , 2002, SIAM J. Comput..

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

[24]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .