Complexity of counting the optimal solutions

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 # ⋅ Π 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 # ⋅ Π k P -classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.