Complexity of Counting the Optimal Solutions

  • Authors:

  • Miki Hermann;Reinhard Pichler

  • Affiliations:

  • LIX (CNRS, UMR 7161), École Polytechnique, Palaiseau, France 91128;Institut für Informationssysteme, Technische Universität Wien, Wien, Austria A-1040

  • Venue:
  • COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
  • Year:
  • 2008

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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茂戮驴 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 #·OptkP and #·OptkP[log n] with k茂戮驴 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.