Eulerian Orientations and Hadamard Codes: A Novel Connection via Counting

Authors Shuai Shao , Zhuxiao Tang



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Author Details

Shuai Shao
  • School of Computer Science and Technology & Hefei National Laboratory, University of Science and Technology of China, Hefei, China
Zhuxiao Tang
  • School of the Gifted Young, University of Science and Technology of China, Hefei, China

Acknowledgements

The first author wants to thank Jin-Yi Cai and Zhiguo Fu for many valuable discussions. The authors want to thank anonymous reviewers for their helpful comments.

Cite As Get BibTex

Shuai Shao and Zhuxiao Tang. Eulerian Orientations and Hadamard Codes: A Novel Connection via Counting. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 86:1-86:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.86

Abstract

We discover a novel connection between two classical mathematical notions, Eulerian orientations and Hadamard codes by studying the counting problem of Eulerian orientations (#EO) with local constraint functions imposed on vertices. We present two special classes of constraint functions and a chain reaction algorithm, and show that the #EO problem defined by each class alone is polynomial-time solvable by the algorithm. These tractable classes of functions are defined inductively, and quite remarkably the base level of these classes is characterized perfectly by the well-known Hadamard code. Thus, we establish a novel connection between counting Eulerian orientations and coding theory. We also prove a #P-hardness result for the #EO problem when constraint functions from the two tractable classes appear together.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Eulerian orientations
  • Hadamard codes
  • Counting problems
  • Tractable classes

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References

  1. Miriam Backens. A new Holant dichotomy inspired by quantum computation. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.16.
  2. Miriam Backens. A complete dichotomy for complex-valued Holant^c. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.12.
  3. Norman Biggs, E Keith Lloyd, and Robin J Wilson. Graph Theory, 1736-1936. Oxford University Press, 1986. Google Scholar
  4. Andrei Bulatov, Martin Dyer, Leslie Ann Goldberg, Markus Jalsenius, and David Richerby. The complexity of weighted Boolean #CSP with mixed signs. Theoretical Computer Science, 410(38-40):3949-3961, 2009. URL: https://doi.org/10.1016/J.TCS.2009.06.003.
  5. Jin-Yi Cai, Zhiguo Fu, and Shuai Shao. Beyond #CSP: A dichotomy for counting weighted eulerian orientations with ars. Information and Computation, 275:104589, 2020. URL: https://doi.org/10.1016/J.IC.2020.104589.
  6. Jin-Yi Cai, Zhiguo Fu, and Shuai Shao. From Holant to quantum entanglement and back. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.22.
  7. Jin-Yi Cai, Zhiguo Fu, and Mingji Xia. Complexity classification of the six-vertex model. Information and Computation, 259:130-141, 2018. URL: https://doi.org/10.1016/J.IC.2018.01.003.
  8. Jin-Yi Cai, Heng Guo, and Tyson Williams. A complete dichotomy rises from the capture of vanishing signatures. SIAM Journal on Computing, 45(5):1671-1728, 2016. URL: https://doi.org/10.1137/15M1049798.
  9. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Dichotomy for Holant^∗ problems of Boolean domain. In Proceedings of the 22nd annual ACM-SIAM symposium on Discrete Algorithms, pages 1714-1728. SIAM, 2011. Google Scholar
  10. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. The complexity of complex weighted Boolean #CSP. Journal of Computer and System Sciences, 80(1):217-236, 2014. URL: https://doi.org/10.1016/J.JCSS.2013.07.003.
  11. Jin-Yi Cai, Pinyan Lu, and Mingji Xia. Dichotomy for real Holant^c problems. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1802-1821. SIAM, 2018. URL: https://doi.org/10.1137/1.9781611975031.118.
  12. Nadia Creignou and Miki Hermann. Complexity of generalized satisfiability counting problems. Information and computation, 125(1):1-12, 1996. URL: https://doi.org/10.1006/INCO.1996.0016.
  13. Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum. The complexity of weighted Boolean #CSP. SIAM Journal on Computing, 38(5):1970-1986, 2009. URL: https://doi.org/10.1137/070690201.
  14. Leonhard Euler. Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, pages 128-140, 1741. Google Scholar
  15. Michel Las Vergnas. On the evaluation at (3, 3) of the Tutte polynomial of a graph. Journal of Combinatorial Theory, Series B, 45(3):367-372, 1988. URL: https://doi.org/10.1016/0095-8956(88)90079-2.
  16. Jiabao Lin and Hanpin Wang. The complexity of Boolean Holant problems with nonnegative weights. SIAM Journal on Computing, 47(3):798-828, 2018. URL: https://doi.org/10.1137/17M113304X.
  17. Boning Meng, Juqiu Wang, and Mingji Xia. P-time algorithms for typical #EO problems. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2410.11557.
  18. Milena Mihail and Peter Winkler. On the number of Eulerian orientations of a graph. Algorithmica, 16(4-5):402-414, 1996. URL: https://doi.org/10.1007/BF01940872.
  19. Linus Pauling. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. Journal of the American Chemical Society, 57(12):2680-2684, 1935. Google Scholar
  20. Franz Rys. Über ein zweidimensionales klassisches Konfigurationsmodell. In Helvetica Physica Acta, volume 36(5), page 537, 1963. Google Scholar
  21. Shuai Shao and Jin-Yi Cai. A dichotomy for real Boolean Holant problems. In Proceedings of the 61st IEEE Annual Symposium on Foundations of Computer Science (FOCS 2020), pages 1091-1102. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00105.
  22. John C Slater. Theory of the transition in KH₂PO₄. The Journal of Chemical Physics, 9(1):16-33, 1941. Google Scholar
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