Abstract:
Following the approach of Hemaspaandra and Vollmer, we can define counting complexity classes #.C for any complexity class C of decision problems. In particular, the classes #.Pi_{k}P with k >= 1 corresponding to all levels of the polynomial hierarchy have thus been studied. However, for a large variety of counting problems arising from optimization problems, a precise complexity classification turns out to be impossible with these classes. In order to remedy this unsatisfactory situation, we introduce a hierarchy of new counting complexity classes #.Opt_{k}P and #.Opt_{k}P[log n] with k >= 1. We prove several important properties of these new classes, like closure properties and the relationship with the #.Pi_{k}P-classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.