Summary: MINIMIZING DNF FORMULAS AND AC0 CIRCUITS GIVEN A TRUTH TABLE
ERIC ALLENDER, LISA HELLERSTEIN, PAUL MCCABE §, TONIANN PITASSI ¶, AND MICHAEL
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 ,
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 (logN) , for some constant > 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(logN) remains open, but we
construct an instance of Min-DNF on which the solution produced by the greedy algorithm is (logN) 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 AC0
d. We show that it is
hard to approximate the size of AC0
d circuits (for large enough d) under cryptographic assumptions.
Key words. Machine learning theory, complexity theory, approximation algorithms, truth table minimization.
AMS subject classifications. 68Q17,68Q32,03D15