The Complexity of Holant Problems over Boolean Domain with Non-negative Weights

Abstract: Holant problem is a general framework to study the computational complexity of counting problems. We prove a complexity dichotomy theorem for Holant problems over Boolean domain with non-negative weights. It is the first complete Holant dichotomy where constraint functions are not necessarily symmetric. Holant problems are indeed read-twice $#$CSPs. Intuitively, some $#$CSPs that are $#$P-hard become tractable when restricting to read-twice instances. To capture them, we introduce the Block-rank-one condition. It turns out that the condition leads to a clear separation. If a function set $\mathcal{F}$ satisfies the condition, then $\mathcal{F}$ is of affine type or product type. Otherwise (a) $\mathrm{Holant}(\mathcal{F})$ is $#$P-hard; or (b) every function in $\mathcal{F}$ is a tensor product of functions of arity at most 2; or (c) $\mathcal{F}$ is transformable to a product type by some real orthogonal matrix. Holographic transformations play an important role in both the hardness proof and the characterization of tractability.