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Holographic Algorithms on Domain Size k > 2

  • Conference paper
Theory and Applications of Models of Computation (TAMC 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7287))

Abstract

An essential problem in the design of holographic algorithms is to decide whether the required signatures can be realized by matchgates under a suitable basis. For domain size two, [1,3] characterized all functions directly realizable as matchgate signatures without a basis transformation, and [7] gave a polynomial time algorithm for the realizability problem for symmetric signatures under basis transformations. We generalize this to arbitrary domain size k. Specifically, we give a polynomial time algorithm for the realizability problem on domain size k ≥ 3. Using this, one can decide whether suitable signatures for a holographic algorithms on domain size k are realizable and if so, to find a suitable linear basis to realize these signatures by an efficient algorithm.

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References

  1. Cai, J.-Y., Choudhary, V.: Some Results on Matchgates and Holographic Algorithms. Int. J. Software and Informatics 1(1), 3–36 (2007)

    MathSciNet  Google Scholar 

  2. Cai, J.-Y., Choudhary, V.: Valiant’s Holant Theorem and matchgate tensors. Theor. Comput. Sci. 384(1), 22–32 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, J.-Y., Choudhary, V., Lu, P.: On the Theory of Matchgate Computations. Theory Comput. Syst. 45(1), 108–132 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, J.-Y., Lu, P.: On Symmetric Signatures in Holographic Algorithms. Theory Comput. Syst. 46(3), 398–415 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cai, J.-Y., Lu, P.: Basis Collapse in Holographic Algorithms. Computational Complexity 17(2), 254–281 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cai, J.-Y., Lu, P.: Holographic algorithms: The power of dimensionality resolved. Theor. Comput. Sci. 410(18), 1618–1628 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai, J.-Y., Lu, P.: Holographic Algorithms: From Art to Science. J. Computer and System Sciences 77, 41–61 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kasteleyn, P.W.: The statistics of dimmers on a lattice. Physica 27, 1209–1225 (1961)

    Article  Google Scholar 

  9. Kasteleyn, P.W.: Graph Theory and Crystal Physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp. 43–110. Academic Press, London (1967)

    Google Scholar 

  10. Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics - an exact result. Philosophical Magazine 6, 1061–1063 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  11. Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM Journal of Computing 31(4), 1229–1254 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Valiant, L.G.: Holographic Algorithms. SIAM J. on Computing 37(5), 1565–1594 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Valiant, L.G.: Accidental Algorithms. In: Proc. 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 509–517 (2006)

    Google Scholar 

  14. Valiant, L.G.: Some Observations on Holographic Algorithms. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 577–590. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  15. Xia, M., Zhang, P., Zhao, W.: Computational complexity of counting problems on 3-regular planar graphs. Theor. Comput. Sci. 384(1), 111–125 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics School of Jilin University, Changchun, Jilin Prov., 130024, China

    Zhiguo Fu

  2. Computer Sciences Department, University of Wisconsin, Madison, WI, 53706, USA

    Jin-Yi Cai

Authors
  1. Zhiguo Fu
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  2. Jin-Yi Cai
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Resource Planning and Generation, Indian Institute of Technology Kanpur, 208016, Kanpur, India

    Manindra Agrawal

  2. Department of Pure Mathematics, Leeds, University of Leeds, LS2 9JT, UK

    S. Barry Cooper

  3. Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, 100190, Beijing, P.R. China

    Angsheng Li

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© 2012 Springer-Verlag Berlin Heidelberg

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Fu, Z., Cai, JY. (2012). Holographic Algorithms on Domain Size k > 2. In: Agrawal, M., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2012. Lecture Notes in Computer Science, vol 7287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29952-0_35

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  • DOI: https://doi.org/10.1007/978-3-642-29952-0_35

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-29951-3

  • Online ISBN: 978-3-642-29952-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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