The Complexity of Planar Boolean #CSP with Complex Weights

  • Heng Guo
  • Tyson Williams
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)

Abstract

We prove a complexity dichotomy theorem for symmetric complex-weighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #P-hard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generalizes a theorem of Cai, Lu, and Xia for the case of real weights. We also obtain a dichotomy theorem for a symmetric arity 4 signature with complex weights in the planar Holant framework, which we use in the proof of our #CSP dichotomy. In particular, we reduce the problem of evaluating the Tutte polynomial of a planar graph at the point (3,3) to counting the number of Eulerian orientations over planar 4-regular graphs to show the latter is #P-hard. This strengthens a theorem by Huang and Lu to the planar setting.

Keywords

Planar Graph Dichotomy Theorem Signature Matrix Constraint Graph Complex Weight 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Guo, H., Williams, T.: The complexity of planar Boolean #CSP with complex weights. CoRR abs/1212.2284 (2012)Google Scholar
  2. 2.
    Valiant, L.G.: The complexity of computing the permanent. TCS 8(2), 189–201 (1979)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Ising, E.: Beitrag zür theorie des ferromagnetismus. Zeitschrift für Physik 31(1), 253–258 (1925)CrossRefGoogle Scholar
  5. 5.
    Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65(3-4), 117–149 (1944)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85(5), 808–816 (1952)MATHCrossRefGoogle Scholar
  7. 7.
    Yang, C.N., Lee, T.D.: Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87(3), 404–409 (1952)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Lee, T.D., Yang, C.N.: Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87(3), 410–419 (1952)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics—an exact result. Philosophical Magazine 6(68), 1061–1063 (1961)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kasteleyn, P.W.: The statistics of dimers on a lattice. Physica 27(12), 1209–1225 (1961)MATHCrossRefGoogle Scholar
  11. 11.
    Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press, London (1982)MATHGoogle Scholar
  12. 12.
    Lieb, E.H., Sokal, A.D.: A general Lee-Yang theorem for one-component and multicomponent ferromagnets. Comm. Math. Phys. 80(2), 153–179 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Welsh, D.: Complexity: Knots, Colourings and Countings. London Mathematical Society Lecture Note Series. Cambridge University Press (1993)Google Scholar
  14. 14.
    Valiant, L.G.: Quantum circuits that can be simulated classically in polynomial time. SIAM J. Comput. 31(4), 1229–1254 (2002)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Valiant, L.G.: Expressiveness of matchgates. TCS 289(1), 457–471 (2002)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Valiant, L.G.: Holographic algorithms. SIAM J. Comput. 37(5), 1565–1594 (2008)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Valiant, L.G.: Accidental algorthims. In: FOCS, pp. 509–517. IEEE Computer Society (2006)Google Scholar
  18. 18.
    Cai, J.Y., Choudhary, V.: Some results on matchgates and holographic algorithms. Int. J. Software and Informatics 1(1), 3–36 (2007)MathSciNetGoogle Scholar
  19. 19.
    Cai, J.Y., Choudhary, V., Lu, P.: On the theory of matchgate computations. Theory of Computing Systems 45(1), 108–132 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Cai, J.Y., Lu, P.: On symmetric signatures in holographic algorithms. Theory of Computing Systems 46(3), 398–415 (2010)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Cai, J.Y., Lu, P.: Holographic algorithms: From art to science. J. Comput. Syst. Sci. 77(1), 41–61 (2011)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Vertigan, D.: The computational complexity of Tutte invariants for planar graphs. SIAM J. Comput. 35(3), 690–712 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Cai, J.Y., Kowalczyk, M.: Spin systems on k-regular graphs with complex edge functions. TCS 461, 2–16 (2012)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    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
  26. 26.
    Cai, J.Y., Lu, P., Xia, M.: Holographic algorithms by Fibonacci gates. Linear Algebra and its Applications 438(2), 690–707 (2013)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Cai, J.Y., Lu, P., Xia, M.: Holographic reduction, interpolation and hardness. Computational Complexity 21(4), 573–604 (2012)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Cai, J.Y., Lu, P., Xia, M.: Holant problems and counting CSP. In: STOC, pp. 715–724. ACM (2009)Google Scholar
  29. 29.
    Cai, J.Y., Lu, P., Xia, M.: Computational complexity of Holant problems. SIAM J. Comput. 40(4), 1101–1132 (2011)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Dyer, M., Goldberg, L.A., Jerrum, M.: The complexity of weighted Boolean CSP. SIAM J. Comput. 38(5), 1970–1986 (2009)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    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
  32. 32.
    Cai, J.Y., Guo, H., Williams, T.: A complete dichotomy rises from the capture of vanishing signatures. CoRR abs/1204.6445 (2012); STOC 2013 (to appear)Google Scholar
  33. 33.
    Las Vergnas, M.: Eulerian circuits of 4-valent graphs imbedded in surfaces. In: Lovász, L., Sós, V.T. (eds.) Algebraic Methods in Graph Theory. Colloq. Math. Soc. János Bolyai, pp. 451–477. North-Holland (1981)Google Scholar
  34. 34.
    Las Vergnas, M.: On the evaluation at (3, 3) of the Tutte polynomial of a graph. J. Comb. Theory, Ser. B 45(3), 367–372 (1988)MATHCrossRefGoogle Scholar
  35. 35.
    Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31(2), 398–427 (2001)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Kowalczyk, M.: Dichotomy theorems for Holant problems. PhD thesis, University of Wisconsin—Madison (2010)Google Scholar
  37. 37.
    Bulatov, A.A., Dalmau, V.: Towards a dichotomy theorem for the counting constraint satisfaction problem. Information and Computation 205(5), 651–678 (2007)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Bulatov, A., Dyer, M., Goldberg, L.A., Jalsenius, M., Richerby, D.: The complexity of weighted boolean #CSP with mixed signs. TCS 410(38-40), 3949–3961 (2009)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Dyer, M., Greenhill, C.: The complexity of counting graph homomorphisms. Random Struct. Algorithms 17(3-4), 260–289 (2000)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Bulatov, A., Grohe, M.: The complexity of partition functions. TCS 348(2), 148–186 (2005)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    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)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    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. LNCS, vol. 6198, pp. 275–286. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Heng Guo
    • 1
  • Tyson Williams
    • 1
  1. 1.University of Wisconsin-MadisonMadisonUSA

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