← 研踪 PaperTrailarXiv:2608.14845 · v2
From Block Orthogonality to Decidability in Complex-Weighted Counting CSP
首次提交 2026-08-15 03:37
版本更新 2026-11-06 23:05
雷达观测 2026-11-09 13:22
摘要
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.
作者备注：68 pages; revised proof and exposition, including a correction; Lean repository linked
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v2 · 2026-11-06 23:05 · 最新观测版本
From Block Orthogonality to Decidability in Complex-Weighted Counting CSP
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.
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