Let $G=(V,E)$ be a simple graph with $|V|=n$. The maximum weight clique problem can be modeled as $\max \sum\limits_{i=1}^n w_ix_i$ subject to $\sum\limits_{i\in S} x_i \leq 1$ for all independent sets $S$ and $x_i \in \{0,1\}$. What is the gap between LP-relaxation and the integral solution?
I think this problem has been studied in the literature but I could not find any reference. Can anyone give me some suggestion?