The Complexity of Symmetric Boolean Parity Holant Problems

(Extended Abstract)
  • Heng Guo
  • Pinyan Lu
  • Leslie G. Valiant
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP-, ⊕P- or #P-complete. Such dichotomy results have been proved for characterizations such as Constraint Satisfaction Problems, and directed and undirected Graph Homomorphism Problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant Problems. These additionally allow for the expression of matching problems, which have had pivotal roles in complexity theory. As our main result we prove the dichotomy theorem that, for the class ⊕P, every set of boolean symmetric Holant signatures of any arities that is not polynomial time computable is ⊕P-complete. The result exploits some special properties of the class ⊕P and characterizes four distinct tractable subclasses within ⊕P. It leaves open the corresponding questions for NP, #P and #kP for k ≠ 2.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arvind, V., Kurur, P.P.: Graph isomorphism is in spp. Inf. Comput. 204(5), 835–852 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beigel, R., Buhrman, H., Fortnow, L.: Np might not be as easy as detecting unique solutions. In: STOC 1998: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pp. 203–208 (1998)Google Scholar
  3. 3.
    Bulatov, A.A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53(1), 66–120 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bulatov, A.A.: The complexity of the counting constraint satisfaction problem. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 646–661. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Cai, J.-Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: A dichotomy theorem. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 275–286. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Cai, J.Y., Huang, S., Lu, P.: From holant to #CSP and back: Dichotomy for holantc problems. arXiv 1004.0803 (2010)Google Scholar
  7. 7.
    Cai, J.Y., Lu, P.: Holographic algorithms: from art to science. In: STOC 2007: Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, pp. 401–410. ACM, New York (2007)CrossRefGoogle Scholar
  8. 8.
    Cai, J.-Y., Lu, P.: Signature theory in holographic algorithms. In: Hong, S.H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 568–579. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Cai, J.Y., Lu, P., Xia, M.: Holographic algorithms by fibonacci gates and holographic reductions for hardness. In: FOCS 2008: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Washington, DC, USA (2008)Google Scholar
  10. 10.
    Cai, J.Y., Lu, P., Xia, M.: A computational proof of complexity of some restricted counting problems. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 138–149. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Cai, J.Y., Lu, P., Xia, M.: Holant problems and counting CSP. In: Mitzenmacher, M. (ed.) STOC, pp. 715–724. ACM, New York (2009)Google Scholar
  12. 12.
    Cai, J.Y., Lu, P., Xia, M.: Holographic algorithms with matchgates capture precisely tractable planar #CSP. In: FOCS 2010: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 427–436 (2010)Google Scholar
  13. 13.
    Cook, M., Bruck, J.: Implementability among predicates. Tech. rep., California Institute of Technology (2005)Google Scholar
  14. 14.
    Creignou, N., Khanna, S., Sudan, M.: Complexity classifications of boolean constraint satisfaction problems. SIAM Monographs on Discrete Mathematics and Applications (2001)Google Scholar
  15. 15.
    Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Inf. Comput. 125(1), 1–12 (1996)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Dodson, C.T.J., Poston, T.: Tensor Geometry. Graduate Texts in Mathematics, vol. 130. Springer, New York (1991)MATHGoogle Scholar
  17. 17.
    Dyer, M.E., Goldberg, L.A., Jerrum, M.: The complexity of weighted boolean #CSP. SIAM J. Comput. 38(5), 1970–1986 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dyer, M.E., Goldberg, L.A., Paterson, M.: On counting homomorphisms to directed acyclic graphs. J. ACM 54(6) (2007)Google Scholar
  19. 19.
    Faben, J.: The complexity of counting solutions to generalised satisfiability problems modulo k. CoRR abs/0809.1836 (2008)Google Scholar
  20. 20.
    Feder, T., Vardi, M.: The computational structure of monotone monadic snp and constraint satisfaction: A study through datalog and group theory. SIAM Journal on Computing 28(1), 57–104 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Goldberg, L.A., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. In: Albers, S., Marion, J.Y. (eds.) STACS. LIPIcs, vol. 3, pp. 493–504. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2009)Google Scholar
  22. 22.
    Guo, H., Huang, S., Lu, P., Xia, M.: The complexity of weighted boolean #csp modulo k. In: Schwentick, T., Dürr, C. (eds.) STACS. LIPIcs, vol. 9, pp. 249–260. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)Google Scholar
  23. 23.
    Kowalczyk, M., Cai, J.Y.: Holant problems for regular graphs with complex edge functions. In: The Proceeding of STACS (2010)Google Scholar
  24. 24.
    Ladner, R.E.: On the structure of polynomial time reducibility. J. ACM 22(1), 155–171 (1975)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Papadimitriou, C.H., Zachos, S.: Two remarks on the power of counting. In: Proceedings of the 6th GI-Conference on Theoretical Computer Science, pp. 269–276 (1982)Google Scholar
  26. 26.
    Schaefer, T.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, p. 226. ACM, New York (1978)Google Scholar
  27. 27.
    Toda, S., Ogiwara, M.: Counting classes are at least as hard as the polynomial-time hierarchy. SIAM J. Comput. 21(2), 316–328 (1992)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47(1), 85–93 (1986)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput. 31(4), 1229–1254 (2002)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Valiant, L.G.: Accidental algorthims. In: FOCS 2006: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 509–517. IEEE Computer Society Press, Washington, DC, USA (2006)CrossRefGoogle Scholar
  32. 32.
    Valiant, L.G.: Holographic algorithms. SIAM J. Comput. 37(5), 1565–1594 (2008)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Valiant, L.G.: Some observations on holographic algorithms. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 577–590. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Heng Guo
    • 1
  • Pinyan Lu
    • 2
  • Leslie G. Valiant
    • 3
  1. 1.University of Wisconsin-MadisonUSA
  2. 2.Microsoft Research AsiaUSA
  3. 3.Harvard UniversityUSA

Personalised recommendations