Guest column: A panorama of counting problems the decision version of which is in P3

Since Valiant's seminal work, where the complexity class #P was defined, much research has been done on counting complexity, from various perspectives. A question that permeates many of these efforts concerns the approximability of counting problems, in particular which of them admit an efficient approximation scheme (fpras). A counting problem (a function from Σ* to N) that admits an fpras must necessarily have an easy way to decide whether the output value is nonzero. Having this in mind, we focus our attention on classes of counting problems, the decision version of which is in P (or in RP). We discuss structural characterizations for classes of such problems under various lenses: Cook and Karp reductions, path counting in non-deterministic Turing machines, approximability and approximation-preserving reductions, easy decision by randomization, descriptive complexity, and interval-size functions. We end up with a rich landscape inside #P, revealing a number of inclusions and separations among complexity classes of easy-to-decide counting problems.

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