In the constructive form of the LLL we bound the expected time using a weight function
$x:\mathcal{A}\rightarrow (0,1)$
that satisfies
$\forall A\in \mathcal{A},\Pr [A] \leq x(A) \prod _{B\in \Gamma^+(A)}(1-x(B))$
and the running time bound is given by $O(\sum_{A\in \mathcal{A}}\frac{x(A)}{1-x(A)})$ expected time
To my understanding there is no nice way to generally compute $x$ or decided it doesn't exist.
I'm interested in finding an approximation of $x$ . I.E a function $x:\mathcal{A}\rightarrow (0,1)$
that satisfies
$\forall A\in \mathcal{A},\pmb{(1-\varepsilon )}\Pr [A] \leq x(A) \prod _{B\in \Gamma^+(A)}(1-x(B))$
for any $0.5>\varepsilon>0 $ . running time should be polynomial in $1/\varepsilon,|\mathcal A|\;\;$and$\;\;\log (1/\min _A(\Pr[A]))$
Is such solution possible, And if so does anyone know of a similar question