Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
SpringerLink
Log in

On Symmetric Signatures in Holographic Algorithms

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

In holographic algorithms, symmetric signatures have been particularly useful. We give a complete characterization of these symmetric signatures over all bases of size 1. These improve previous results by Cai and Choudhary (ICALP 2006, vol. 4051, pp. 703–714, 2006) where only symmetric signatures over the Hadamard basis (special basis of size 1) were obtained. In particular, we give a complete list of Boolean symmetric signatures over bases of size 1.

It is an open problem whether signatures over bases of higher dimensions are strictly more powerful. The recent result by Valiant (FOCS 2006, pp. 509–517, 2006) seems to suggest that bases of size 2 might be indeed more powerful than bases of size 1. This result is with regard to a restrictive counting version of #SAT called #Pl-Rtw-Mon-3CNF. It is known that the problem is #P-hard, and its mod 2 version is ⊕P-hard. Yet its mod 7 version is solvable in polynomial time by holographic algorithms. This was ac complished by a suitable symmetric signature over a basis of size 2. We show that the same unexpected holographic algorithm can be realized over a basis of size 1. Furthermore we prove that 7 is the only modulus for which such an “accidental algorithm” exists.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bubley, R., Dyer, M.: Graph orientations with no sink and an approximation for a hard case of #SAT. In: ACM SODA, pp. 248–257 (1997)

  2. Cai, J.-Y., Choudhary, V.: Valiant’s Holant theorem and matchgate tensors. In: Proceedings of TAMC 2006. Lecture Notes in Computer Science, vol. 3959, pp. 248–261. Theor. Comput. Sci. 384(1), 22–32 (2007). Also available as ECCC TR05-118

  3. Cai, J.-Y., Choudhary, V.: Some results on matchgates and holographic algorithms. In: Proceedings of ICALP 2006, Part I. Lecture Notes in Computer Science, vol. 4051, pp. 703–714. Springer, Berlin (2007). Int. J. Softw. Inf. 1(1), 3–36

    Google Scholar 

  4. Cai, J.-Y., Choudhary, V., Lu, P.: On the theory of matchgate computations. In: Proceedings of CCC’07: Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, pp. 305–318. IEEE Computer Society, Washington (2007)

    Chapter  Google Scholar 

  5. Cai, J.-Y., Lu, P.: On symmetric signatures in holographic algorithms. In: Thomas, W., Weil, P. (eds.) Proceedings of STACS. Lecture Notes in Computer Science, vol. 4393, pp. 429–440. Springer, Berlin (2007)

    Google Scholar 

  6. Cai, J.-Y., Lu, P.: Holographic algorithms: from art to science. In: Proceedings of STOC’07: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 401–410. ACM, New York (2007)

    Chapter  Google Scholar 

  7. Cai, J.-Y., Lu, P.: Bases collapse in holographic algorithms. In: Proceedings of CCC ’07: Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, pp. 292–304. IEEE Computer Society, Washington (2007). Comput. Complexity 17(2), 254–281 (2008)

    Chapter  Google Scholar 

  8. Cai, J.-Y., Lu, P.: Holographic algorithms: the power of dimensionality resolved. In: Arge, L., Cachin, C., Jurdzinski, T. et al. (eds.) Proceedings of ICALP. Lecture Notes in Computer Science, vol. 4596, pp. 631–642. Springer, Berlin (2007)

    Google Scholar 

  9. Cai, J.-Y., Lu, P.: On block-wise symmetric signatures for matchgates. In: Csuhaj-Varjú, E., Ésik, Z. (eds.) Proceedings of FCT. Lecture Notes in Computer Science, vol. 4639, pp. 187–198. Springer, Berlin (2007)

    Google Scholar 

  10. Cai, J.-Y., Lu, P.: Holographic algorithms with unsymmetric signatures. In: Proceedings of SODA ’08: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 54–63. SIAM, Philadelphia (2008)

    Google Scholar 

  11. Cai, J.-Y., Lu, P.: Signature theory in holographic algorithms. In: Hong, S.H., Nagamochi, H., Fukunaga, T. (eds.) Proceedings of ISAAC. Lecture Notes in Computer Science, vol. 5369, pp. 568–579. Springer, Berlin (2008)

    Google Scholar 

  12. Hunt, H.B., Marathe, M.V., Radhakrishnan, V., Stearns, R.E.: The complexity of planar counting problems. SIAM J. Comput. 27(4), 1142–1167 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hunt, H.B. III, Stearns, R.E.: The complexity of very simple Boolean formulas with applications. SIAM J. Comput. 19(1), 44–70 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  15. 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 

  16. Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11, 329–343 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  17. Murota, K.: Matrices and Matroids for Systems Analysis. Springer, Berlin (2000)

    MATH  Google Scholar 

  18. Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics—an exact result. Philos. Mag. 6, 1061–1063 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  19. Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput. 31(4), 1229–1254 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Valiant, L.G.: Expressiveness of matchgates. Theor. Comput. Sci. 281(1), 457–471 (2002). See also Theor. Comput. Sci. 299, 795 (2003)

    Article  MathSciNet  Google Scholar 

  21. Valiant, L.G.: Holographic algorithms (extended abstract). In: Proc. 45th IEEE Symposium on Foundations of Computer Science, pp. 306–315 (2004). A more detailed version appeared in Electronic Colloquium on Computational Complexity Report TR05-099

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

Download references

Author information

Authors and Affiliations

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

    Jin-Yi Cai

  2. Department of Computer Science and Technology, Tsinghua University, Beijing, 100084, P.R. China

    Pinyan Lu

Authors
  1. Jin-Yi Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pinyan Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jin-Yi Cai.

Additional information

A preliminary version of this paper appeared in STACS 2007 [5].

J.-Y. Cai was supported by NSF CCF-0208013 and CCF-0511679.

P. Lu was supported by the National Natural Science Foundation of China Grant 60553001 and the National Basic Research Program of China Grants 2007CB807900, 2007CB807901.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cai, JY., Lu, P. On Symmetric Signatures in Holographic Algorithms. Theory Comput Syst 46, 398–415 (2010). https://doi.org/10.1007/s00224-009-9229-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-009-9229-z

Keywords

Search

Navigation