Skip to main content
SpringerLink
Log in

The Complexity of Planar Boolean #CSP with Complex Weights

  • Conference paper
Automata, Languages, and Programming (ICALP 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7965))

Included in the following conference series:

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.

Full version with proofs available at [1].

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Guo, H., Williams, T.: The complexity of planar Boolean #CSP with complex weights. CoRR abs/1212.2284 (2012)

    Google Scholar 

  2. Valiant, L.G.: The complexity of computing the permanent. TCS 8(2), 189–201 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  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. Ising, E.: Beitrag zür theorie des ferromagnetismus. Zeitschrift für Physik 31(1), 253–258 (1925)

    Article  Google Scholar 

  5. Onsager, L.: Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65(3-4), 117–149 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  6. Yang, C.N.: The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 85(5), 808–816 (1952)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics—an exact result. Philosophical Magazine 6(68), 1061–1063 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kasteleyn, P.W.: The statistics of dimers on a lattice. Physica 27(12), 1209–1225 (1961)

    Article  MATH  Google Scholar 

  11. Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press, London (1982)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  13. Welsh, D.: Complexity: Knots, Colourings and Countings. London Mathematical Society Lecture Note Series. Cambridge University Press (1993)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Valiant, L.G.: Expressiveness of matchgates. TCS 289(1), 457–471 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Valiant, L.G.: Holographic algorithms. SIAM J. Comput. 37(5), 1565–1594 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Valiant, L.G.: Accidental algorthims. In: FOCS, pp. 509–517. IEEE Computer Society (2006)

    Google Scholar 

  18. Cai, J.Y., Choudhary, V.: Some results on matchgates and holographic algorithms. Int. J. Software and Informatics 1(1), 3–36 (2007)

    MathSciNet  Google Scholar 

  19. Cai, J.Y., Choudhary, V., Lu, P.: On the theory of matchgate computations. Theory of Computing Systems 45(1), 108–132 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Cai, J.Y., Lu, P.: On symmetric signatures in holographic algorithms. Theory of Computing Systems 46(3), 398–415 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cai, J.Y., Lu, P.: Holographic algorithms: From art to science. J. Comput. Syst. Sci. 77(1), 41–61 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Vertigan, D.: The computational complexity of Tutte invariants for planar graphs. SIAM J. Comput. 35(3), 690–712 (2005)

    Article  MathSciNet  Google Scholar 

  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. Cai, J.Y., Kowalczyk, M.: Spin systems on k-regular graphs with complex edge functions. TCS 461, 2–16 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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. Cai, J.Y., Lu, P., Xia, M.: Holographic algorithms by Fibonacci gates. Linear Algebra and its Applications 438(2), 690–707 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Cai, J.Y., Lu, P., Xia, M.: Holographic reduction, interpolation and hardness. Computational Complexity 21(4), 573–604 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Cai, J.Y., Lu, P., Xia, M.: Holant problems and counting CSP. In: STOC, pp. 715–724. ACM (2009)

    Google Scholar 

  29. Cai, J.Y., Lu, P., Xia, M.: Computational complexity of Holant problems. SIAM J. Comput. 40(4), 1101–1132 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dyer, M., Goldberg, L.A., Jerrum, M.: The complexity of weighted Boolean CSP. SIAM J. Comput. 38(5), 1970–1986 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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)

    Article  MATH  Google Scholar 

  35. Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31(2), 398–427 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kowalczyk, M.: Dichotomy theorems for Holant problems. PhD thesis, University of Wisconsin—Madison (2010)

    Google Scholar 

  37. Bulatov, A.A., Dalmau, V.: Towards a dichotomy theorem for the counting constraint satisfaction problem. Information and Computation 205(5), 651–678 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  39. Dyer, M., Greenhill, C.: The complexity of counting graph homomorphisms. Random Struct. Algorithms 17(3-4), 260–289 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  40. Bulatov, A., Grohe, M.: The complexity of partition functions. TCS 348(2), 148–186 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Wisconsin-Madison, Madison, WI, USA

    Heng Guo & Tyson Williams

Authors
  1. Heng Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tyson Williams
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Fedor V. Fomin

  2. Faculty of Computing, University of Latvia, Raina bulv. 19, 1586, Riga, Latvia

    Rūsiņš Freivalds

  3. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Marta Kwiatkowska

  4. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel

    David Peleg

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Guo, H., Williams, T. (2013). The Complexity of Planar Boolean #CSP with Complex Weights. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_44

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-39206-1_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39205-4

  • Online ISBN: 978-3-642-39206-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation