Subtractive Reductions and Complete Problems for Counting Complexity Classes

  • Arnaud Durand
  • Miki Hermann
  • Phokion G. Kolaitis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)


We introduce and investigate a new type of reductions between counting problems, which we call subtractive reductions. We show that the main counting complexity classes #P, #NP, as well as all higher counting complexity classes #·ΠP k, k≥2, are closed under subtractive reductions. We then pursue problems that are complete for these classes via subtractive reductions. We focus on the class #NP (which is the same as the class #·coNP) and show that it contains natural complete problems via subtractive reductions, such as the problem of counting the minimal models of a Boolean formula in conjunctive normal form and the problem of counting the cardinality of the set of minimal solutions of a homogeneous system of linear Diophantine inequalities.


Conjunctive Normal Form Truth Assignment Boolean Formula Complete Problem Counting Problem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Arnaud Durand
    • 1
  • Miki Hermann
    • 2
  • Phokion G. Kolaitis
    • 3
  1. 1.LACL, Dept. of Computer ScienceUniversité Paris 12CréteilFrance
  2. 2.LORIA (CNRS)Vandœuvre-lés-NancyFrance
  3. 3.Computer Science DeptUniversity of CaliforniaSanta CruzUSA

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