Zeros and approximations of Holant polynomials on the complex plane

Abstract

We present fully polynomial time approximation schemesfor a broad class of Holant problems with complex edge weights, whichwe call Holant polynomials. We transform these problems into partitionfunctions of abstract combinatorial structures known as polymersin statistical physics. Our method involves establishing zero-free regionsfor the partition functions of polymer models and using the mostsignificant terms of the cluster expansion to approximate them.Results of our technique include new approximation and sampling algorithmsfor a diverse class of Holant polynomials in the low-temperatureregime (i.e. small external field) and approximation algorithms for generalHolant problems with small signature weights. Additionally, wegive randomised approximation and sampling algorithms with fasterrunning times for more restrictive classes. Finally, we improve theknown zero-free regions for a perfect matching polynomial.