4
$\begingroup$

Suppose I'm given a CNF formula with $m$ clauses (and $k$ literals in each clause), with a total of $n$ variables in the formula, where each variable is in at most $c$ clauses, along with a satisfying assignment to this formula. I change the value of each variable to the other possible value (from the one in the given assignment) with constant probability $p \leq \frac{1}{2}$ independently of the other variables. I want to show that with probability at least $1-exp\left(-\frac{p^2n}{2c^2}\right)$, the new assignment satisfies at least $\left(1-\left(c+1\right)p\right)m$ clauses.

It seems I should use here martingales (Azuma's inequality) or large deviation inequalities, however I didn't manage to get any meaningful results using these methods, so that may not be the case.

$\endgroup$
  • $\begingroup$ Is it clear that the expected fraction of violated clauses is indeed $p(c+1)$ ? $\endgroup$ – Suresh Venkat Jun 4 '11 at 19:15
  • $\begingroup$ @Suresh: That doesn't seem correct - if that were indeed the expected fraction of violated clauses, it would mean that most of our assignments are concentrated 'below' the expected value -- at least for me, this makes no sense. Also, for the 'worst' case, where there are $m=c$ clauses, each variable is in exactly $c$ clauses, and each change in any variable's value causes all $c$ of those clauses to be violated, we get that the expected fraction of violated clauses is p, and this does not match with your observation. $\endgroup$ – shay Jun 5 '11 at 8:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.