We classify Boolean Holant problems with a supplied binary disequality signature and an arbitrary fixed finite set of algebraic complex-valued signatures. Each problem is polynomial-time computable or $\#\mathsf{P}$-hard, and the tractability criterion is decidable. The proof combines arity reduction with the growth of the available binary transfer group and uses stable pairs to recover the structure of higher-arity signatures. We also prove a dichotomy for the disequality-edge model when the signature set contains an endpoint-nondegenerate irreducible eight-vertex signature, resolving the corresponding arity-four cases in the main proof.
A Boolean Holant problem evaluates a graph partition function specified by local signatures at its vertices. We prove a complexity dichotomy for arbitrary finite sets of algebraic complex-valued signatures on general graphs: an explicit, decidable criterion characterizes the polynomial-time cases, and every remaining case is $\#\mathsf{P}$-hard under polynomial-time Turing reductions. The proof resolves the all-even case by constructing signatures with nonzero endpoints and all remaining support on the central layer, then classifying them according to half their arity. Binary transfer groups and matching-product reconstruction provide the main tools in the small- and large-height cases, respectively.
In a landmark JACM paper recognized with the 2021 Gödel Prize, Cai and Chen established a complexity dichotomy for counting CSPs over arbitrary finite domains with algebraic complex weights. Their tractability criterion consists of three conditions on the infinite family of functions obtained from CSP instances by summing out some variables: the Block Orthogonality and Type Partition conditions, and a Mal'tsev polymorphism condition on support and row-equivalence relations. They left open whether this criterion is decidable from the finite set of constraint functions. We resolve this question by giving a total exact algorithm for every $q$-element domain ($q\ge 1$), with algebraic weights encoded exactly. We also prove that the Block Orthogonality condition implies the other two conditions, reducing the tractability criterion to a single structural property. We obtain similar decidability and structural results for Lin's tractability criterion for $\#\mathrm{CSP}^{d}$, where each variable occurs a multiple of $d$ times. Our method first shows that every failure of Block Orthogonality can be witnessed by a generated function of arity at most $q^2+1$. At each such arity, we reduce checking infinitely many generated functions to finitely many exact algebraic tests, using tensor products and a pinned extension of Young's all-instance isomorphism theorem. For the structural implication, we use powers of generated functions to control complex cancellation and construct functions whose supports encode row equivalence. These constructions yield the Type Partition and Mal'tsev polymorphism conditions.
We prove Friedgut's conjecture (J. Amer. Math. Soc. 12 (1999)) that every increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have size bounded independently of the dimension and the bias. For $0<p\le 1/2$, every increasing Boolean function with total resampling influence at most $K$ is $\varepsilon$-close under $μ_p^n$ to a monotone DNF of width $\exp(O((K+1)^2/\varepsilon^2))$, from which the full conjecture follows for all $p\in(0,1)$. We build on Hatami's pseudo-junta theorem (Ann. of Math. 176 (2012)), whose approximator need not be increasing. Our randomized shifts produce an increasing approximator that is measurable with respect to a controlled refinement of Hatami's adaptive partition. Retaining only short positive certificates yields the desired narrow DNF.