Complexity theory of real functions
Complexity theory of real functions
Approximating the permanent of graphs with large factors
Theoretical Computer Science
Finite fields
Complexity and real computation
Complexity and real computation
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
The Complexity of Counting in Sparse, Regular, and Planar Graphs
SIAM Journal on Computing
Quantum Circuits That Can Be Simulated Classically in Polynomial Time
SIAM Journal on Computing
Towards a Dichotomy Theorem for the Counting Constraint Satisfaction Problem
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Corrigendum: the complexity of counting graph homomorphisms
Random Structures & Algorithms
A dichotomy theorem for constraint satisfaction problems on a 3-element set
Journal of the ACM (JACM)
The complexity of partition functions
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
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
The Complexity of Weighted Boolean CSP
SIAM Journal on Computing
On Symmetric Signatures in Holographic Algorithms
Theory of Computing Systems - Special Issue: Theoretical Aspects of Computer Science; Guest Editors: Wolgang Thomas and Pascal Weil
Graph homomorphisms with complex values: a dichotomy theorem
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Holographic algorithms: From art to science
Journal of Computer and System Sciences
On counting homomorphisms to directed acyclic graphs
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Guest column: complexity dichotomies of counting problems
ACM SIGACT News
The complexity of planar boolean #CSP with complex weights
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
The complexity of complex weighted Boolean #CSP
Journal of Computer and System Sciences
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We propose and explore a novel alternative framework to study the complexity of counting problems, called Holant problems. Compared to counting constraint satisfaction problems (#CSP), it is a refinement with a more explicit role for the constraint functions. Both graph homomorphism and #CSP can be viewed as special cases of Holant problems. We prove complexity dichotomy theorems in this framework. Our dichotomy theorems apply to local constraint functions, which are symmetric functions on Boolean input variables and evaluate to arbitrary real or complex values. We discover surprising tractable subclasses of counting problems, which could not easily be specified in the #CSP framework. When all unary functions are assumed to be free ($\mathrm{Holant}^*$ problems), the tractable ones consist of functions that are degenerate, or of arity at most two, or holographic transformations of Fibonacci gates. When only two special unary functions, the constant zero and constant one functions, are assumed to be free ($\mathrm{Holant}^c$ problems), we further identify three special families of tractable cases. Then we prove that all other cases are #P-hard. The main technical tool we use and develop is holographic reductions. Another technical tool used in combination with holographic reductions is polynomial interpolations.