The complexity of elementary algebra and geometry
Journal of Computer and System Sciences
On the complexity of H-coloring
Journal of Combinatorial Theory Series B
Complexity of generalized satisfiability counting problems
Information and Computation
Complexity and real computation
Complexity and real computation
The complexity of counting graph homomorphisms
Proceedings of the ninth international conference on on Random structures and 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
Holographic Algorithms (Extended Abstract)
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
The complexity of partition functions
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
Towards a dichotomy theorem for the counting constraint satisfaction problem
Information and Computation
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)
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
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
Graph homomorphisms with complex values: a dichotomy theorem
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Spin systems on graphs with complex edge functions and specified degree regularities
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Spin systems on k-regular graphs with complex edge functions
Theoretical Computer Science
Partition functions on k-regular graphs with {0,1}-vertex assignments and real edge functions
Theoretical Computer Science
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We explore a computational approach to proving the intractability of certain counting problems. These problems can be described in various ways, and they include concrete problems such as counting the number of vertex covers or independent sets for 3-regular graphs. The high level principle of our approach is algebraic, which provides sufficient conditions for interpolation to succeed. Another algebraic component is holographic reductions. We then analyze in detail polynomial maps on R^2 induced by some combinatorial constructions. These maps define sufficiently complicated dynamics of R^2 that we can only analyze them computationally. In this paper we use both numerical computation (as intuitive guidance) and symbolic computation (as proof theoretic verification) to derive that a certain collection of combinatorial constructions, in myriad combinations, fulfills the algebraic requirements of proving #P-hardness. The final result is a dichotomy theorem for a class of counting problems. This includes a class of generic holant problems with an arbitrary real valued edge signature over (2,3)-regular undirected graphs. In particular, it includes all partition functions with 0-1 vertex assignments and an arbitrary real valued edge function over all 3-regular undirected graphs.