Minimizing Disjunctive Normal Form Formulas and $AC^0$ Circuits Given a Truth Table

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 disjunctive normal form (DNF), and Min-Circuit (also called the minimum circuit size problem (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 [Some NP-Complete Set Covering Problems, manuscript, 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 $(\log N)^{\gamma}$, for some constant $\gamma0$, assuming that NP is not contained in quasi-polynomial 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 $\Omega(\log N)$ larger than optimal. Finally, we turn to the question of approximating circuit size for slightly more general classes of circuits. DNF formulas are depth-two circuits of AND and OR gates. Depth-$d$ circuits are denoted by $AC^0_d$. We show that it is hard to approximate the size of $AC^0_d$ circuits (for large enough $d$) under cryptographic assumptions.