<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2011-06-04T19:15:38.207Z
Is it clear that the expected fraction of violated clauses is indeed $p(c+1)$  ?
Suresh Venkat
CC BY-SA 3.0
2011-06-05T08:04:42.943Z
@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.
shay
<co.combinatorics><pr.probability>
Adding noise to an assignment
shay
# Adding noise to an assignment

(Asked by: username_0 on 2011-06-04)

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.

**Tags:** co.combinatorics, pr.probability
> **Comments:**
> **username_1**: Is it clear that the expected fraction of violated clauses is indeed $p(c+1)$  ?  
> **username_0**: @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.  

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## 0 answers

