Complexity of counting the optimal solutions

  • Authors:

  • Miki Hermann;Reinhard Pichler

  • Affiliations:

  • LIX (CNRS, UMR 7161), École Polytechnique, 91128 Palaiseau cedex, France;Institut für Informationssysteme, Technische Universität Wien, Favoritenstrasse 9-11, A-1040 Wien, Austria

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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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 #@?@P"kP 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"kP and #@?Opt"kP[logn] with k=1. We prove several important properties of these new classes, like closure properties and the relationship with the #@?@P"kP-classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.