Holant Problems for Regular Graphs with Complex Edge Functions

Michael Kowalczyk & Jin-Yi Cai
We prove a complexity dichotomy theorem for Holant Problems on $3$-regular graphs with an arbitrary complex-valued edge function. Three new techniques are introduced: (1) higher dimensional iterations in interpolation; (2) Eigenvalue Shifted Pairs, which allow us to prove that a pair of combinatorial gadgets \emph{in combination} succeed in proving \#P-hardness; and (3) algebraic symmetrization, which significantly lowers the \emph{symbolic complexity} of the proof for computational complexity. With \emph{holographic reductions} the classification theorem also applies to...