 
Summary: MINIMIZING DNF FORMULAS AND AC0 CIRCUITS GIVEN A TRUTH TABLE
ERIC ALLENDER, LISA HELLERSTEIN, PAUL MCCABE §, TONIANN PITASSI ¶, AND MICHAEL
SAKS
Abstract. For circuit classes R, the fundamental computational problem MinR asks for the minimum Rsize of
a Boolean function presented as a truth table. Prominent examples of this problem include MinDNF, which asks
whether a given Boolean function presented as a truth table has a kterm DNF, and MinCircuit (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 MinDNF is NPcomplete. It is significantly simpler than the known reduction of Masek [30],
which is from CircuitSAT. We then give a more complex reduction, yielding the result that MinDNF 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 MinDNF can be approximated to within a factor of o(logN) remains open, but we
construct an instance of MinDNF 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
