Complexity of Counting the Optimal Solutions

Abstract

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 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 -classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.KeywordsTuring MachineComplexity ClassVertex CoverConjunctive Normal FormPropositional FormulaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.