NP is as easy as detecting unique solutions
Theoretical Computer Science
Counting classes are at least as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Complexity of generalized satisfiability counting problems
Information and Computation
NP might not be as easy as detecting unique solutions
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
Complexity classifications of boolean constraint satisfaction problems
Complexity classifications of boolean constraint satisfaction problems
Quantum Circuits That Can Be Simulated Classically in Polynomial Time
SIAM Journal on Computing
Two remarks on the power of counting
Proceedings of the 6th GI-Conference on Theoretical Computer Science
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
A dichotomy theorem for constraint satisfaction problems on a 3-element set
Journal of the ACM (JACM)
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Holographic algorithms: from art to science
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On counting homomorphisms to directed acyclic graphs
Journal of the ACM (JACM)
SIAM Journal on Computing
The Complexity of the Counting Constraint Satisfaction Problem
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Holographic Algorithms by Fibonacci Gates and Holographic Reductions for Hardness
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Signature Theory in Holographic Algorithms
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
Holant problems and counting CSP
Proceedings of the forty-first annual ACM symposium on Theory of computing
A Computational Proof of Complexity of Some Restricted Counting Problems
TAMC '09 Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation
The Complexity of Weighted Boolean CSP
SIAM Journal on Computing
Information and Computation
Graph homomorphisms with complex values: a dichotomy theorem
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Holographic Algorithms with Matchgates Capture Precisely Tractable Planar_#CSP
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Some observations on holographic algorithms
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Gadgets and anti-gadgets leading to a complexity dichotomy
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
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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 #P-complete. Such dichotomy results have been proved for characterizations such as Constraint Satisfaction Problems, and directed and undirected Graph Homomorphism Problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant Problems. These additionally allow for the expression of matching problems, which have 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 ⊕P-complete. The result exploits some special properties of the class ⊕P and characterizes four distinct tractable subclasses within ⊕P. It leaves open the corresponding questions for NP, #P and #kP for k ≠ 2.