Posted by: atri | March 23, 2009

Lecture 25: l-wise independent sources

In today’s lecture, we defined the notion of an \ell-wise independent sources (as I mentioned in the lecture, these are generally called k-wise independent sources but for us k is already taken). We also saw how given an \ell-wise independent sources, we can come up with a 1-\frac{1}{2^{\ell}}-approximation algorithm for the the MAXE\ellSAT problem. (In the problem, we are given a bunch of clauses where each clause has exactly \ell literals. The goal is to come up with an assignment to the variables that satisfies as many clauses as possible.)

Next lecture, we will see that the dual of the BCH_{2,\log{n},\ell+1} code is an \ell-wise independent source. By the bounds on the dimension of these codes, this means that we can get an \ell-wise independent source of size O\left(n^{\lfloor \frac{\ell}{2}\rfloor}\right). Further, as these codes are linear codes, each codeword can be generated in time O(n^2). This implies, that we have an 1-2^{-\ell}-approximation algorithm for MAXE\ellSAT that runs in time O\left(n^{2+\lfloor \frac{\ell}{2}\rfloor}\right).

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