Amplifying lower bounds by means of self-reducibility
Journal of the ACM (JACM)
Uniform derandomization from pathetic lower bounds
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory
Journal of Computer and System Sciences
The complexity of Boolean formula minimization
Journal of Computer and System Sciences
Exponential lower bounds for AC0-Frege imply superpolynomial frege lower bounds
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Arithmetic circuits: The chasm at depth four gets wider
Theoretical Computer Science
On the gap between ess(f) and cnf_size(f)
Discrete Applied Mathematics
Compact DSOP and Partial DSOP Forms
Theory of Computing Systems
Hi-index | 0.00 |
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.