On the complexity of H-coloring
Journal of Combinatorial Theory Series B
Polynomial-time approximation algorithms for the Ising model
SIAM Journal on Computing
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
Complexity of generalized satisfiability counting problems
Information and Computation
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
The complexity of counting graph homomorphisms (extended abstract)
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
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
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Holographic Algorithms (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
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
Holant problems and counting CSP
Proceedings of the forty-first annual ACM symposium on Theory of computing
An approximation trichotomy for Boolean #CSP
Journal of Computer and System Sciences
Proceedings of the forty-second ACM symposium on Theory of computing
Graph homomorphisms with complex values: a dichotomy theorem
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
A Decidable Dichotomy Theorem on Directed Graph Homomorphisms with Non-negative Weights
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Codes on graphs: normal realizations
IEEE Transactions on Information Theory
Guest column: complexity dichotomies of counting problems
ACM SIGACT News
Gadgets and anti-gadgets leading to a complexity dichotomy
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
The expressibility of functions on the boolean domain, with applications to counting CSPs
Journal of the ACM (JACM)
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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 output 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 over the complex domain, we discover surprising new tractable classes, which are associated with isotropic vectors, i.e., a (non-zero) vector whose inner product with itself is zero.