Suppose we have a linear progamming of vertex packing for a hypergraph (V,E), with size $n = \sum_{e \in E} |e|$. We introduce a variable $x_v$ for each vertex $v \in V$. The integral version is stated as follows:
$$\max \sum_{v \in V} x_v $$ $$s.t. \ \sum_{v: v \in e} x_v \le 1, \forall e \in E$$ $$\ \ \ \ x_v \in \{0,1\}$$
We have anonther version of this LP with non-integer constraints as follows:
$$\max \sum_{v \in V} x_v $$ $$s.t. \ \sum_{v: v \in e} x_v \le 1, \forall e \in E$$ $$\ \ \ \ x_v \in \{0, \log_N 2, \log_N 3, \cdots, 1\}$$
Let the optimal solution for these two LPs as $OPT_1$ and $OPT_2$.
Is there any good upper bound for $\frac{OPT_1}{OPT_2}$ in terms of n and N?
