TY - GEN

T1 - Inimizing DNF formulas and ACd0 circuits given a truth table

AU - Allender, Eric

AU - Hellerstein, Lisa

AU - McCabe, Paul

AU - Pitassi, Toniann

AU - Saks, Michael

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

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U2 - 10.1109/CCC.2006.27

DO - 10.1109/CCC.2006.27

M3 - Conference contribution

AN - SCOPUS:34247522421

SN - 0769525962

SN - 9780769525969

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 237

EP - 251

BT - Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006

T2 - 21st Annual IEEE Conference on Computational Complexity, CCC 2006

Y2 - 16 July 2006 through 20 July 2006

ER -