Abstract
I would like to report on some significant progress in the study of the exact complexity of counting problems. Specifically I will describe the classification program of counting complexity of locally specified problems. This classification program is advanced in three interrelated frameworks: Graph Homomorphisms, Counting CSP, and Holant Problems. In each formulation, complexity dichotomy theorems have been achieved which classify every problem in a given class to be either solvable in polynomial time or #P-hard.
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