Complexity of Counting the Optimal Solutions

  • Miki Hermann
  • Reinhard Pichler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5092)


Following the approach of Hemaspaandra and Vollmer, we can define counting complexity classes #\(\cdot\mathcal{C}\) for any complexity class Open image in new window of decision problems. In particular, the classes Open image in new window 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 Open image in new window -classes. Moreover, we establish the completeness of several natural counting complexity problems for these new classes.


Turing Machine Complexity Class Vertex Cover Conjunctive Normal Form Propositional Formula 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Miki Hermann
    • 1
  • Reinhard Pichler
    • 2
  1. 1.LIX (CNRS, UMR 7161)École PolytechniquePalaiseauFrance
  2. 2.Institut für InformationssystemeTechnische Universität WienWienAustria

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