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On the Complexity of #CSP^d

Author Jiabao Lin

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Author Details

Jiabao Lin
  • Shanghai University of Finance and Economics, China

Funding

  • Lin, Jiabao: This work is supported by Science and Technology Innovation 2030 –“New Generation of Artificial Intelligence” Major Project No.(2018AAA0100903), NSFC grant 61932002, Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities.

Acknowledgements

I would like to thank anonymous referees for their insightful comments and suggestions.

Cite As Get BibTex

Jiabao Lin. On the Complexity of #CSP^d. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 40:1-40:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ITCS.2021.40

Abstract

Counting CSP^d is the counting constraint satisfaction problem (#CSP in short) restricted to the instances where every variable occurs a multiple of d times. This paper revisits tractable structures in #CSP and gives a complexity classification theorem for #CSP^d with algebraic complex weights. The result unifies affine functions (stabilizer states in quantum information theory) and related variants such as the local affine functions, the discovery of which leads to all the recent progress on the complexity of Holant problems.
The Holant is a framework that generalizes counting CSP. In the literature on Holant problems, weighted constraints are often expressed as tensors (vectors) such that projections and linear transformations help analyze the structure. This paper gives an example showing that different classes of tensors distinguished by these algebraic operations may share the same closure property under tensor product and contraction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Constraint satisfaction problem
  • counting problems
  • Holant
  • complexity dichotomy

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References

  1. Miriam Backens. A new holant dichotomy inspired by quantum computation. In 44th International Colloquium on Automata, Languages,and Programming, ICALP 2017, pages 16:1-16:14, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.16.
  2. Miriam Backens. A complete dichotomy for complex-valued Holant^c. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 12:1-12:14, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.12.
  3. Andrei A. Bulatov. The complexity of the counting constraint satisfaction problem. J. ACM, 60(5):34, 2013. URL: https://doi.org/10.1145/2528400.
  4. Jin-Yi Cai and Xi Chen. Complexity of counting CSP with complex weights. J. ACM, 64(3):19:1-19:39, 2017. URL: https://doi.org/10.1145/2822891.
  5. Jin-Yi Cai, Xi Chen, and Pinyan Lu. Graph homomorphisms with complex values: A dichotomy theorem. SIAM J. Comput., 42(3):934-1029, 2013. URL: https://doi.org/10.1137/110840194.
  6. Jin-Yi Cai, Xi Chen, and Pinyan Lu. Nonnegative weighted #CSP: An effective complexity dichotomy. SIAM J. Comput., 45(6):2177-2198, 2016. URL: https://doi.org/10.1137/15m1032314.
  7. Jin-Yi Cai, Zhiguo Fu, Heng Guo, and Tyson Williams. A Holant dichotomy: Is the FKT algorithm universal? In IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1259-1276, 2015. URL: https://doi.org/10.1109/focs.2015.81.
  8. Jin-Yi Cai, Zhiguo Fu, and Shuai Shao. From Holant to quantum entanglement and back. In 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), pages 22:1-22:16, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.22.
  9. Jin-Yi Cai, Heng Guo, and Tyson Williams. The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems. In 55th IEEE Annual Symposium on Foundations of Computer Science, pages 601-610, 2014. URL: https://doi.org/10.1109/FOCS.2014.70.
  10. Jin-Yi Cai, Heng Guo, and Tyson Williams. A complete dichotomy rises from the capture of vanishing signatures. SIAM J. Comput., 45(5):1671-1728, 2016. URL: https://doi.org/10.1137/15m1049798.
  11. Jin-Yi Cai, Heng Guo, and Tyson Williams. Clifford gates in the holant framework. Theor. Comput. Sci., 745:163-171, 2018. URL: https://doi.org/10.1016/j.tcs.2018.06.010.
  12. Jin-Yi Cai, Sangxia Huang, and Pinyan Lu. From Holant to #CSP and back: Dichotomy for Holant^c problems. Algorithmica, 64(3):511-533, 2012. URL: https://doi.org/10.1007/s00453-012-9626-6.
  13. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Holant problems and counting CSP. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pages 715-724, 2009. URL: https://doi.org/10.1145/1536414.1536511.
  14. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Dichotomy for Holant^* problems with domain size 3. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1278-1295, 2013. URL: https://doi.org/10.1137/1.9781611973105.93.
  15. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. The complexity of complex weighted Boolean #CSP. J. Comput. Syst. Sci., 80(1):217-236, 2014. URL: https://doi.org/10.1016/j.jcss.2013.07.003.
  16. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Dichotomy for real Holant^c problems. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1802-1821, 2018. URL: https://doi.org/10.1137/1.9781611975031.118.
  17. Martin E. Dyer, Leslie A. Goldberg, and Mark Jerrum. The complexity of weighted Boolean #CSP. SIAM J. Comput., 38(5):1970-1986, 2009. URL: https://doi.org/10.1137/070690201.
  18. Martin E. Dyer and David Richerby. An effective dichotomy for the counting constraint satisfaction problem. SIAM J. Comput., 42(3):1245-1274, 2013. URL: https://doi.org/10.1137/100811258.
  19. Leslie A. Goldberg, Martin Grohe, Mark Jerrum, and Marc Thurley. A complexity dichotomy for partition functions with mixed signs. SIAM J. Comput., 39(7):3336-3402, 2010. URL: https://doi.org/10.1137/090757496.
  20. Sangxia Huang and Pinyan Lu. A dichotomy for real weighted Holant problems. Computational Complexity, 25(1):255-304, 2016. URL: https://doi.org/10.1007/s00037-015-0118-3.
  21. Jiabao Lin and Hanpin Wang. The complexity of Boolean Holant problems with nonnegative weights. SIAM J. Comput., 47(3):798-828, 2018. URL: https://doi.org/10.1137/17M113304X.
  22. László Lovász. Large Networks and Graph Limits, volume 60 of Colloquium Publications. American Mathematical Society, 2012. URL: http://www.ams.org/bookstore-getitem/item=COLL-60.
  23. Shuai Shao and Jin-Yi Cai. A dichotomy for real Boolean Holant problems. To appear at FOCS, 2020. URL: http://arxiv.org/abs/2005.07906.
  24. Leslie G. Valiant. Holographic algorithms. SIAM J. Comput., 37(5):1565-1594, 2008. URL: https://doi.org/10.1137/070682575.
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