Abstract
In this paper we investigate the problem of approximating the fraction of truth assignments that satisfy a Boolean formula with some restricted form of DNF under distributions with limited independence between random variables. LetF be a DNF formula onn variables withm clauses in which each literal appears at most once. We prove that ifD is [k logm]-wise independent, then |Pr D [F]-Pr U [F]| ≤\((\frac{1}{2})^{\Omega (k)} \), whereU denotes the uniform distribution and Pr D [F] denotes the probability thatF is satisfied by a truth assignment chosen according to distributionD (similarly for Pr U [F]). Using the result, we also derive the following: For formulas satisfying the restriction described above and for any constantc, there exists a probability distributionD, with size polynomial in logn andm, such that |Pr D [F] - Pr U [F]| ≤c holds.
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