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Approximability of Minimum AND-Circuits

Authors: Jan Arpe and Bodo Manthey

Published in: Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

Abstract

Given a set of monomials, the {sc Minimum AND-Circuit} problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size. We prove that the problem is not polynomial time approximable within a factor of less than $1.0051$ unless $mathsf{P} = mathsf{NP}$, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of $1.278$. For the general problem, we achieve an approximation ratio of $d-3/2$, where $d$ is the degree of the largest monomial. In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we reveal connections between the {sc Minimum AND-Circuit} problem and several problems from different areas.

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Jan Arpe and Bodo Manthey. Approximability of Minimum AND-Circuits. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)

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@InProceedings{arpe_et_al:DagSemProc.06111.4,
  author =	{Arpe, Jan and Manthey, Bodo},
  title =	{{Approximability of Minimum AND-Circuits}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--21},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.4},
  URN =		{urn:nbn:de:0030-drops-6039},
  doi =		{10.4230/DagSemProc.06111.4},
  annote =	{Keywords: Optimization problems, approximability, automated circuit design}
}

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Approximability of Minimum AND-Circuits

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