Dichotomy for Holant∗ Problems on the Boolean Domain

Holant problems are a general framework to study counting problems. Both counting constraint satisfaction problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ ( F ) , where F is a set of constraint functions on Boolean variables and taking c...

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Bibliographic Details
Published in:Theory of computing systems Vol. 64; no. 8; pp. 1362 - 1391
Main Authors: Cai, Jin-Yi, Lu, Pinyan, Xia, Mingji
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2020
Springer Nature B.V
Subjects:
Algorithms
Boolean
Boolean algebra
Complex variables
Complexity
Computer Science
Homomorphisms
Polynomials
Theorems
Theory of Computation
P-hardness
Constraint satisfaction problems
Dichotomy theorems
Edge coloring models
Holant problems
Polynomial time algorithms
ISSN:1432-4350, 1433-0490
Online Access:Get full text
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Summary:Holant problems are a general framework to study counting problems. Both counting constraint satisfaction problems (#CSP) and graph homomorphisms are special cases. We prove a complexity dichotomy theorem for Holant ∗ ( F ) , where F is a set of constraint functions on Boolean variables and taking complex values. The constraint functions need not be symmetric functions. We identify four classes of problems which are polynomial time computable; all other problems are proved to be #P-hard. The main proof technique and indeed the formulation of the theorem use holographic algorithms and reductions. By considering these counting problems with the broader scope that allows complex-valued constraint functions, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (non-zero) vector whose dot product with itself is zero.
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-020-09983-8