Skip to main content
SpringerLink
Log in

Local holographic transformations: tractability and hardness

  • Research Article
  • Published:
Frontiers of Computer Science Aims and scope Submit manuscript

This article has been updated

Abstract

Local holographic transformations were introduced by Cai et al., and local affine functions, an extra tractable class, were derived by it in #CSP2. In the present paper, we not only generalize local affine functions to #CSPd for general d, but also give new tractable classes by combining local holographic transformations with global holographic transformations. Moreover, we show how to use local holographic transformations to prove hardness. This is of independent interests in the complexity classification of counting problems.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

  • 13 August 2022

    Incorrect cover date was used, instead of 2022 it should be 2023.

References

  1. Valiant L G. Quantum circuits that can be simulated classically in polynomial time. SIAM Journal on Computing, 2002, 31(4): 1229–1254

    Article  MathSciNet  Google Scholar 

  2. Valiant L G. Holographic algorithms. SIAM Journal on Computing, 2008, 37(5): 1565–1594

    Article  MathSciNet  Google Scholar 

  3. Kasteleyn P W. The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. Physica, 1961, 27(12): 1209–1225

    Article  Google Scholar 

  4. Harary F. Graph Theory and Theoretical Physics. London: Academic Press, 1967: 43–110

    MATH  Google Scholar 

  5. Temperley H N V, Fisher M E. Dimer problem in statistical mechanicsan exact result. The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics, 1961, 6(68): 1061–1063

    Article  Google Scholar 

  6. Valiant L G. Accidental algorthims. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science. 2006, 509–517

  7. Bulatov A, Dyer M, Goldberg L A, Jalsenius M, Jerrum M, Richerby D. The complexity of weighted and unweighted #CSP. Journal of Computer and System Sciences, 2012, 78(2): 681–688

    Article  MathSciNet  Google Scholar 

  8. Backens M. A complete dichotomy for complex-valued holantc. In: Proceedings of the 45th International Colloquium on Automata, Languages, and Programming. 2018, 1–14

  9. Cai J Y, Chen X, Lu P. Graph homomorphisms with complex values: a dichotomy theorem. SIAM Journal on Computing, 2013, 42(3): 924–1029

    Article  MathSciNet  Google Scholar 

  10. Cai J Y, Guo H, Williams T. A complete dichotomy rises from the capture of vanishing signatures. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing. 2013, 635–644

  11. Cai J Y, Lu P, Xia M. The complexity of complex weighted Boolean #CSP. Journal of Computer and System Sciences, 2014, 80(1): 217–236

    Article  MathSciNet  Google Scholar 

  12. Cai J Y, Fu Z, Guo H, Williams T. A holant dichotomy: is the FKT algorithm universal? In: Proceedings of the IEEE 56th Annual Symposium on Foundations of Computer Science. 2015, 1259–1276

  13. Cai J Y, Chen X, Lu P. Nonnegative weighted #CSP: an effective complexity dichotomy. SIAM Journal on Computing, 2016, 45(6): 2177–2198

    Article  MathSciNet  Google Scholar 

  14. Cai J Y, Chen X. Complexity of counting CSP with complex weights. Journal of the ACM, 2017, 64(3): 19

    Article  MathSciNet  Google Scholar 

  15. Cai J Y, Fu Z. Holographic algorithm with matchgates is universal for planar #CSP over Boolean domain. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing. 2017, 842–855

  16. Lin J, Wang H. The complexity of Boolean Holant problems with nonnegative weights. SIAM Journal on Computing, 2018, 47(3): 798–828

    Article  MathSciNet  Google Scholar 

  17. Shao S, Cai J Y. A dichotomy for real Boolean holant problems. In: Proceedings of the 61st IEEE Annual Symposium on Foundations of Computer Science. 2020, 1091–1102

  18. Huang S, Lu P. A dichotomy for real weighted holant problems. Computational Complexity, 2016, 25(1): 255–304

    Article  MathSciNet  Google Scholar 

  19. Cai J Y, Lu P, Xia M. Dichotomy for real holantc problems. In: Proceedings of 2018 Annual ACM-SIAM Symposium on Discrete Algorithms. 2018, 1802–1821

  20. Lin J B. The complexity of counting CSPd. Theory of Computing Systems, 2021, 65(6): 1–13

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 61872076) and the Natural Science Foundation of Jilin Province (20200201161JC).

Author information

Authors and Affiliations

  1. School of Information Science and Technology & KLAS, Northeast Normal University, Changchun, 130117, China

    Peng Yang & Zhiguo Fu

Authors
  1. Peng Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhiguo Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiguo Fu.

Additional information

Peng Yang is currently a PhD student in the School of Information Science and Technology, Northeast Normal University, China. His currently research interests include computational complexity of counting problems.

Zhiguo Fu received the PhD degree in the school of Mathematics from Jilin University, China. He is an associate professor and doctoral supervisor at Northeast Normal University, China. His research interests lie in theoretical computer science.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, P., Fu, Z. Local holographic transformations: tractability and hardness. Front. Comput. Sci. 17, 172401 (2023). https://doi.org/10.1007/s11704-022-1231-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11704-022-1231-5

Keywords

Search

Navigation