On the Connection between Interval Size Functions and Path Counting

We investigate the complexity of hard counting problems that belong to the class #P but have easy decision version; several well-known problems such as # Perfect Matchings , # DNFSat share this property. We focus on classes of such problems which emerged through two disparate approaches: one taken by Hemaspaandra et al. [1] who defined classes of functions that count the size of intervals of ordered strings, and one followed by Kiayias et al. [2] who defined the class TotP, consisting of functions that count the total number of paths of NP computations. We provide inclusion and separation relations between TotP and interval size counting classes, by means of new classes that we define in this work. Our results imply that many known #P-complete problems with easy decision are contained in the classes defined in [1]--but are unlikely to be complete for these classes under certain types of reductions. We also define a new class of interval size functions which strictly contains FP and is strictly contained in TotP under reasonable complexity-theoretic assumptions. We show that this new class contains some hard counting problems.