The Complexity of Counting Functions with Easy Decision Version

  • Aris Pagourtzis
  • Stathis Zachos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

We investigate the complexity of counting problems that belong to the complexity class #P and have an easy decision version. These problems constitute the class #PE which has some well-known representatives such as #Perfect Matchings, #DNF-Sat, and NonNegative Permanent. An important property of these problems is that they are all #P-complete, in the Cook sense, while they cannot be #P-complete in the Karp sense unless P = NP.

We study these problems in respect to the complexity class TotP, which contains functions that count the number of all paths of a PNTM. We first compare TotP to #P and #PE and show that FPTotP#PE#P and that the inclusions are proper unless P = NP.

We then show that several natural #PE problems — including the ones mentioned above — belong to TotP. Moreover, we prove that TotP is exactly the Karp closure of self-reducible functions of #PE. Therefore, all these problems share a remarkable structural property: for each of them there exists a polynomial-time nondeterministic Turing machine which has as many computation paths as the output value.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Aris Pagourtzis
    • 1
  • Stathis Zachos
    • 1
    • 2
  1. 1.Department of Computer Science, School of ECENational Technical University of AthensGreece
  2. 2.CIS DepartmentBrooklyn College, CUNY 

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