Inimizing DNF formulas and ACd0 circuits given a truth table

Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, Michael Saks

Research output: Chapter in Book/Report/Conference proceedingConference contribution

15 Scopus citations

Abstract

For circuit classes R, the fundamental computational problem Min-R asks for the minimum R-size of a Boolean function presented as a truth table. Prominent examples of this problem include Min-DNF, which asks whether a given Boolean function presented as a truth table has a k-term DNF, and Min-Circuit (also called MCSP), which asks whether a Boolean function presented as a truth table has a size k Boolean circuit. We present a new reduction proving that Min-DNF is NP-complete. It is significantly simpler than the known reduction of Masek [31], which is from Circuit-SAT. We then give a more complex reduction, yielding the result that Min-DNF cannot be approximated to within a factor smaller than (log N)γ, for some constant γ > 0, assuming that NP is not contained in quasipolynomial time. The standard greedy algorithm for Set Cover is often used in practice to approximate Min-DNF. The question of whether Min-DNF can be approximated to within a factor of o(log N) remains open, but we construct an instance of Min-DNF on which the solution produced by the greedy algorithm is Ω(log N) larger than optimal. Finally, we extend known hardness results for Min-TCd0 to obtain new hardness results for Min-ACd0, under cryptographic assumptions.

Original languageEnglish (US)
Title of host publicationProceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006
Pages237-251
Number of pages15
DOIs
StatePublished - 2006
Event21st Annual IEEE Conference on Computational Complexity, CCC 2006 - Prague, Czech Republic
Duration: Jul 16 2006Jul 20 2006

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
Volume2006
ISSN (Print)1093-0159

Other

Other21st Annual IEEE Conference on Computational Complexity, CCC 2006
Country/TerritoryCzech Republic
CityPrague
Period7/16/067/20/06

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Computational Mathematics

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