Complexity of Counting the Optimal Solutions

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 #·ΠkP 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 #·OptkP[logn] with k ≥ 1. We prove several important properties of these new classes, like closure properties and the relationship with the #·ΠkP-classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.