Completeness Results for Counting Problems with Easy Decision.

Counting problems with easy decision are the only ones among problems in complexity class #P that are likely to be (randomly) approximable, under the assumption RP not equal NP. TotP is a subclass of #P that contains many of these problems. TotP and #P share some complete problems under Cook reductions, the approximability of which does not extend to all problems in these classes (if RP not equal = NP); the reason is that such reductions do not preserve the function value. Therefore Cook reductions do not seem useful in obtaining (in) approximability results for counting problems in TotP and #P. On the other hand, the existence of TotP-complete problems (apart from the generic one) under stronger reductions that preserve the function value has remained an open question thus far. In this paper we present the first such problems, the definitions of which are related to satisfiability of Boolean circuits and formulas. We also discuss implications of our results to the complexity and approximability of counting problems in general.