Approximation Algorithm for the Unique Coverage Problem
Suppose there are $m$ elements and $n$ sets (this is the meaning of your parameter $n$). The algorithm divides the set of elements into $\log n$ classes, so one of these classes, say class $i$, contains at least $m/\log n$ elements. Lemma 3 shows that if you pick each set with probability $1/2^i$, then in expectation you uniquely cover a $1/e^2$ fraction of the elements in class $i$. Thus, you uniquely cover at least $m/(e^2\log n)$ elements in expectation. On the other hand, the optimal cover uniquely covers at most $m$ elements. Thus the approximation ratio of this algorithm is at least $1/(e^2\log n)$.