Abstract
We present fully polynomial time approximation schemes for a broad class of Holant problems with complex edge weights, which we call Holant polynomials. We transform these problems into partition functions of abstract combinatorial structures known as polymers in statistical physics. Our method involves establishing zero-free regions for the partition functions of polymer models and using the most significant terms of the cluster expansion to approximate them. Results of our technique include new approximation and sampling algorithms for a diverse class of Holant polynomials in the low-temperature regime (i.e. small external field) and approximation algorithms for general Holant problems with small signature weights. Additionally, we give randomised approximation and sampling algorithms with faster running times for more restrictive classes. Finally, we improve the known zero-free regions for a perfect matching polynomial.
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19 October 2022
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Acknowledgements
The authors would like to thank Heng Guo, Guus Regts as well as an anonymous referee for helpful comments on drafts of this work. Andreas Göbel was funded by the project PAGES (project No. 467516565) of the German Research Foundation (DFG).
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Casel, K., Fischbeck, P., Friedrich, T. et al. Zeros and approximations of Holant polynomials on the complex plane. comput. complex. 31, 11 (2022). https://doi.org/10.1007/s00037-022-00226-5
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DOI: https://doi.org/10.1007/s00037-022-00226-5