Complexity of Counting the Optimal Solutions

Following the approach of Hemaspaandra and Vollmer, we can define counting complexity classes #$\cdot\mathcal{C}$ for any complexity class of decision problems. In particular, the classes with ki¾? 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 ki¾? 1. We prove several important properties of these new classes, like closure properties and the relationship with the -classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.

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