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Dichotomy for Holant Problems of 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 comp ..."
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Cited by 13 (4 self)
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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
The Complexity of Symmetric Boolean Parity Holant Problems (Extended Abstract)
"... Abstract. For certain subclasses of NP, ⊕P or #P characterized by local constraints, it is known that if there exist any problems that are not polynomial time computable within that subclass, then those problems are NP, ⊕P or #Pcomplete. Such dichotomy results have been proved for characterizatio ..."
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Cited by 12 (3 self)
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had pivotal roles in complexity theory. As our main result we prove the dichotomy theorem that, for the class ⊕P, every set of boolean symmetric Holant signatures of any arities that is not polynomial time computable is ⊕Pcomplete. The result exploits some special properties of the class ⊕P
The Complexity of Planar Boolean #CSP with Complex Weights
"... We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates. This generali ..."
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Cited by 6 (3 self)
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We prove a complexity dichotomy theorem for symmetric complexweighted Boolean #CSP when the constraint graph of the input must be planar. The problems that are #Phard over general graphs but tractable over planar graphs are precisely those with a holographic reduction to matchgates
A Holant Dichotomy: Is the FKT Algorithm Universal?
"... We prove a complexity dichotomy for complexweighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specically to answer the question: Is the FKT algorithm under a holographic transformation [38] a universal strategy to obtain polynom ..."
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Cited by 1 (1 self)
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We prove a complexity dichotomy for complexweighted Holant problems with an arbitrary set of symmetric constraint functions on Boolean variables. This dichotomy is specically to answer the question: Is the FKT algorithm under a holographic transformation [38] a universal strategy to obtain
A Complete Dichotomy Rises from the Capture of Vanishing Signatures
"... We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complexvalued symmetric constraint functions F on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions taking values in a field of characteristi ..."
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Cited by 10 (6 self)
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We prove a complexity dichotomy theorem for Holant problems over an arbitrary set of complexvalued symmetric constraint functions F on Boolean variables. This extends and unifies all previous dichotomies for Holant problems on symmetric constraint functions taking values in a field
From Holant To #CSP And Back: Dichotomy For Holant c Problems
"... We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted Hcolorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results fro ..."
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Cited by 10 (6 self)
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We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted Hcolorings. We consider these problems for general complex valued functions that take boolean inputs. We show that results
Dichotomy for Holant* Problems with a Function on Domain Size 3
, 2012
"... Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the Boolean domain. In this paper, we give the first dichotomy ..."
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Holant problems are a general framework to study the algorithmic complexity of counting problems. Both counting constraint satisfaction problems and graph homomorphisms are special cases. All previous results of Holant problems are over the Boolean domain. In this paper, we give the first dichotomy
Holographic Algorithms with Matchgates Capture Precisely Tractable Planar #CSP
"... Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a un ..."
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Cited by 17 (7 self)
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in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary realvalued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs