@article{CaselFischbeckFriedrichetal.2022,
  author    = {Casel, Katrin and Fischbeck, Philipp and Friedrich, Tobias and G{\"o}bel, Andreas and Lagodzinski, J. A. Gregor},
  title     = {Zeros and approximations of Holant polynomials on the complex plane},
  series = {Computational complexity : CC},
  volume    = {31},
  journal   = {Computational complexity : CC},
  number    = {2},
  publisher = {Springer},
  address   = {Basel},
  issn      = {1016-3328},
  doi       = {10.1007/s00037-022-00226-5},
  pages     = {52},
  year      = {2022},
  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.},
  language  = {en}
}
@article{GoebelLagodzinskiSeidel2021,
  author    = {G{\"o}bel, Andreas and Lagodzinski, Gregor J. A. and Seidel, Karen},
  title     = {Counting homomorphisms to trees modulo a prime},
  series = {ACM transactions on computation theory : TOCT / Association for Computing Machinery},
  volume    = {13},
  journal   = {ACM transactions on computation theory : TOCT / Association for Computing Machinery},
  number    = {3},
  publisher = {Association for Computing Machinery},
  address   = {New York},
  issn      = {1942-3454},
  doi       = {10.1145/3460958},
  pages     = {1 -- 33},
  year      = {2021},
  abstract  = {Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of \#(p) HOMSTOH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies. <br /> Our main result states that for every tree H and every prime p the problem \#pHOMSTOH is either polynomial time computable or \#P-p-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of \#pHOMSTOH are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p. These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes p.},
  language  = {en}
}