k-polynomial time approximation algorithm for set cover (k = max size of subsets)
Let $A = \{ i : x_i \geq 1/k \}$. By construction, $|A| \leq k \sum_i x_i \leq k \mathrm{OPT}$. The set $A$ is a set cover, since if $S_j = \{i_1,\ldots,i_\ell\}$ (where $\ell \leq k$) then $x_{i_1} + \cdots + x_{i_\ell} \geq 1$, and so $\max(x_{i_1},\ldots,x_{i_\ell}) \geq 1/\ell \geq 1/k$.