Hardness of Minimizing and Learning DNF Expressions

We study the problem of finding the minimum size DNF formula for a function f : {0, 1}d → {0, 1} given its truth table. We show that unless NP ⊆ DTIME(npoly(log n) ), there is no polynomial time algorithm that approximates this problem to within factor d1−ε where ε > 0 is an arbitrarily small constant. Our result essentially matches the known O(d) approximation for the problem. We also study weak learnability of small size DNF formulas. We show that assuming NP ⊆ RP, for arbitrarily small constant ε > 0 and any fixed positive integer t, a two term DNF cannot be PAC-learnt in polynomial time by a t term DNF to within 1 2 + ε accuracy. Under the same complexity assumption, we show that for arbitrarily small constants µ, ε > 0 and any fixed positive integer t, an AND function (i.e. a single term DNF) cannot be PAC-learnt in polynomial time under adversarial µ-noise by a t-CNF to within 1 2 + ε accuracy.

• Type: Conference
• Year: 2008
• Where: FOCS
• Authors: Subhash Khot, Rishi Saket
• Keywords: Arbitrarily Small, FOCS 2008, Size Dnf Formulas, Term Dnf, Theoretical Computer Science