Minimizing DNF Formulas and AC^0_d Circuits Given a Truth Table
Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, Michael Saks · 2006
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 (1979), 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)/sup /spl Upsi//, for some constant /spl Upsi/ > 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 /spl Omega/(log N) larger than optimal. Finally, we extend known hardness results for Min-TC/sup 0//sub d/ to obtain new hardness results for Min-AC/sup 0//sub d/, under cryptographic assumptions.