Holographic Algorithms Beyond Matchgates

  • Jin-Yi Cai
  • Heng Guo
  • Tyson Williams
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

Abstract

Holographic algorithms based on matchgates were introduced by Valiant. These algorithms run in polynomial-time and are intrinsically for planar problems. We introduce two new families of holographic algorithms, which work over general, i.e., not necessarily planar, graphs. The two underlying families of constraint functions are of the affine and product types. These play the role of Kasteleyn’s algorithm for counting planar perfect matchings. The new algorithms are obtained by transforming a problem to one of these two families by holographic reductions. We present a polynomial-time algorithm to decide if a given counting problem has a holographic algorithm using these constraint families. When the constraints are symmetric, we give a polynomial-time decision procedure in the size of the succinct presentation of symmetric constraint functions. This procedure shows that the recent dichotomy theorem for Holant problems with symmetric constraints is polynomial-time decidable.

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References

  1. 1.
    Bulatov, A., Dyer, M., Goldberg, L.A., Jalsenius, M., Richerby, D.: The complexity of weighted Boolean #CSP with mixed signs. Theor. Comput. Sci. 410(38-40), 3949–3961 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bulatov, A., Grohe, M.: The complexity of partition functions. Theor. Comput. Sci. 348(2), 148–186 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bulatov, A.A.: The complexity of the counting constraint satisfaction problem. J. ACM 60(5), 34:1–34:41 (2013)Google Scholar
  4. 4.
    Bulatov, A.A., Dalmau, V.: Towards a dichotomy theorem for the counting constraint satisfaction problem. Inform. and Comput. 205(5), 651–678 (2007)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cai, J.-Y., Chen, X.: A decidable dichotomy theorem on directed graph homomorphisms with non-negative weights. In: FOCS, pp. 437–446. IEEE Computer Society (2010)Google Scholar
  6. 6.
    Cai, J.-Y., Chen, X.: Complexity of counting CSP with complex weights. In: STOC, pp. 909–920. ACM (2012), arXiv:1111.2384Google Scholar
  7. 7.
    Cai, J.-Y., Chen, X., Lu, P.: Non-negatively weighted #CSP: An effective complexity dichotomy. In: IEEE Conference on Computational Complexity, pp. 45–54. IEEE Computer Society (2011)Google Scholar
  8. 8.
    Cai, J.-Y., Chen, X., Lu, P.: Graph homomorphisms with complex values: A dichotomy theorem. SIAM J. Comput. 42(3), 924–1029 (2013)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Cai, J.-Y., Choudhary, V., Lu, P.: On the theory of matchgate computations. Theory of Computing Systems 45(1), 108–132 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Cai, J.-Y., Gorenstein, A.: Matchgates revisited. Theory of Computing 10(868), 4001–4030 (2014)Google Scholar
  11. 11.
    Cai, J.-Y., Guo, H., Williams, T.: A complete dichotomy rises from the capture of vanishing signatures (extended abstract). In: STOC, pp. 635–644. ACM (2013), arXiv:1204.6445Google Scholar
  12. 12.
    Cai, J.-Y., Guo, H., Williams, T.: Holographic algorithms beyond matchgates. CoRR, abs/1307.7430 (2013)Google Scholar
  13. 13.
    Cai, J.-Y., Huang, S., Lu, P.: From Holant to #CSP and back: Dichotomy for Holantc problems. Algorithmica 64(3), 511–533 (2012)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Cai, J.-Y., Kowalczyk, M.: Spin systems on k-regular graphs with complex edge functions. Theor. Comput. Sci. 461, 2–16 (2012)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Cai, J.-Y., Kowalczyk, M., Williams, T.: Gadgets and anti-gadgets leading to a complexity dichotomy. In: ITCS, pp. 452–467. ACM (2012)Google Scholar
  16. 16.
    Cai, J.-Y., Lu, P.: Holographic algorithms with unsymmetric signatures. In: SODA, pp. 54–63. Society for Industrial and Applied Mathematics (2008)Google Scholar
  17. 17.
    Cai, J.-Y., Lu, P.: Holographic algorithms: From art to science. J. Comput. Syst. Sci. 77(1), 41–61 (2011)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Cai, J.-Y., Lu, P.: Signature theory in holographic algorithms. Algorithmica 61(4), 779–816 (2011)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Cai, J.-Y., Lu, P., Xia, M.: Holographic algorithms with matchgates capture precisely tractable planar #CSP. In: FOCS, pp. 427–436. IEEE Computer Society (2010)Google Scholar
  20. 20.
    Cai, J.-Y., Lu, P., Xia, M.: Computational complexity of Holant problems. SIAM J. Comput. 40(4), 1101–1132 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Cai, J.-Y., Lu, P., Xia, M.: Dichotomy for Holant* problems of Boolean domain. In: SODA, pp. 1714–1728. SIAM (2011)Google Scholar
  22. 22.
    Cai, J.-Y., Lu, P., Xia, M.: Holographic reduction, interpolation and hardness. Computational Complexity 21(4), 573–604 (2012)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Cai, J.-Y., Lu, P., Xia, M.: Dichotomy for Holant* problems with domain size 3. In: SODA, pp. 1278–1295. SIAM (2013)Google Scholar
  24. 24.
    Cai, J.-Y., Lu, P., Xia, M.: Holographic algorithms by Fibonacci gates. Linear Algebra and its Applications 438(2), 690–707 (2013)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Cai, J.-Y., Lu, P., Xia, M.: The complexity of complex weighted Boolean #CSP. J. Comput. System Sci. 80(1), 217–236 (2014)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Dyer, M., Goldberg, L.A., Jerrum, M.: The complexity of weighted Boolean #CSP. SIAM J. Comput. 38(5), 1970–1986 (2009)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Dyer, M., Goldberg, L.A., Paterson, M.: On counting homomorphisms to directed acyclic graphs. J. ACM 54(6) (2007)Google Scholar
  28. 28.
    Dyer, M., Greenhill, C.: The complexity of counting graph homomorphisms. Random Struct. Algorithms 17(3-4), 260–289 (2000)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Dyer, M., Richerby, D.: An effective dichotomy for the counting constraint satisfaction problem. SIAM J. Comput. 42(3), 1245–1274 (2013)CrossRefMATHGoogle Scholar
  30. 30.
    Goldberg, L.A., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. SIAM J. Comput. 39(7), 3336–3402 (2010)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Guo, H., Huang, S., Lu, P., Xia, M.: The complexity of weighted Boolean #CSP modulo k. In: STACS, pp. 249–260. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)Google Scholar
  32. 32.
    Guo, H., Williams, T.: The complexity of planar Boolean #CSP with complex weights. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 516–527. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  33. 33.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Comb. Theory Ser. B 48(1), 92–110 (1990)CrossRefMATHGoogle Scholar
  34. 34.
    Huang, S., Lu, P.: A dichotomy for real weighted Holant problems. In: IEEE Conference on Computational Complexity, pp. 96–106. IEEE Computer Society (2012)Google Scholar
  35. 35.
    Kasteleyn, P.W.: The statistics of dimers on a lattice. Physica 27(12), 1209–1225 (1961)CrossRefMATHGoogle Scholar
  36. 36.
    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
  37. 37.
    Kayal, N.: Affine projections of polynomials: Extended abstract. In: STOC, pp. 643–662. ACM (2012)Google Scholar
  38. 38.
    Lovász, L.: Operations with structures. Acta Math. Hung. 18(3-4), 321–328 (1967)CrossRefMATHGoogle Scholar
  39. 39.
    Mulmuley, K.D., Sohoni, M.: Geometric complexity theory I: An approach to the P vs. NP and related problems. SIAM J. Comput. 31(2), 496–526 (2001)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Temperley, H.N.V.: Michael E. Fisher. Dimer problem in statistical mechanics—an exact result. Philosophical Magazine 6(68), 1061–1063 (1961)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Valiant, L.G.: Expressiveness of matchgates. Theor. Comput. Sci. 289(1), 457–471 (2002)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput. 31(4), 1229–1254 (2002)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Valiant, L.G.: Accidental algorthims. In: FOCS, pp. 509–517. IEEE Computer Society (2006)Google Scholar
  44. 44.
    Valiant, L.G.: Holographic algorithms. SIAM J. Comput. 37(5), 1565–1594 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Heng Guo
    • 1
  • Tyson Williams
    • 1
  1. 1.University of Wisconsin–MadisonMadisonUSA

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