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The Complexity of Counting Functions with Easy Decision Version

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Mathematical Foundations of Computer Science 2006 (MFCS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,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 FP⊆TotP⊆#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.

Research supported in part by grant EΔ285, under the ΠENEΔ 2003 programme of the General Secretariat of Research and Technology of Greece.

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Author information

Authors and Affiliations

  1. Department of Computer Science, School of ECE, National Technical University of Athens, Greece

    Aris Pagourtzis & Stathis Zachos

  2. CIS Department, Brooklyn College, CUNY,  

    Stathis Zachos

Authors
  1. Aris Pagourtzis
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  2. Stathis Zachos
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Comenius University, Mlynská dolina, 84248, Bratislava, Slovakia

    Rastislav Královič

  2. Institute of Informatics, University of Warsaw, Poland

    Paweł Urzyczyn

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Pagourtzis, A., Zachos, S. (2006). The Complexity of Counting Functions with Easy Decision Version. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_64

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  • DOI: https://doi.org/10.1007/11821069_64

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37791-7

  • Online ISBN: 978-3-540-37793-1

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