Theory of Computing Systems

, Volume 46, Issue 3, pp 398–415 | Cite as

On Symmetric Signatures in Holographic Algorithms



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.


Holographic algorithms Matchgates and signatures Symmetric signatures 0-1 signatures Realizability 


  1. 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) Google Scholar
  2. 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 Google Scholar
  3. 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. 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) CrossRefGoogle Scholar
  5. 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. 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) CrossRefGoogle Scholar
  7. 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) CrossRefGoogle Scholar
  8. 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. 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. 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. 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. 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) MATHCrossRefMathSciNetGoogle Scholar
  13. 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) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kasteleyn, P.W.: The statistics of dimers on a lattice. Physica 27, 1209–1225 (1961) CrossRefGoogle Scholar
  15. 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. 16.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11, 329–343 (1982) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Murota, K.: Matrices and Matroids for Systems Analysis. Springer, Berlin (2000) MATHGoogle Scholar
  18. 18.
    Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics—an exact result. Philos. Mag. 6, 1061–1063 (1961) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput. 31(4), 1229–1254 (2002) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Valiant, L.G.: Expressiveness of matchgates. Theor. Comput. Sci. 281(1), 457–471 (2002). See also Theor. Comput. Sci. 299, 795 (2003) CrossRefMathSciNetGoogle Scholar
  21. 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 Google Scholar
  22. 22.
    Valiant, L.G.: Accidental algorithms. In: Proc. 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 509–517 (2006) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA
  2. 2.Department of Computer Science and TechnologyTsinghua UniversityBeijingP.R. China

Personalised recommendations