Before revision:  Extending a result of Khuller, Moss, and Naor for the Budgeted Coverage problem, Sviridenko showed that the greedy algorithm combined with guessing 3 most profitable elements of an optimal solution has approximation ratio \alpha = 1-\frac{1 \approx 0.632 . We show that just 2 guesses sufficeto achieve ratio \alpha  , 1 guesssuffices to achieve ratio 0.899 \alpha \approx 0.568, while ratio 0.68\alpha \approx 0.43 can be achieved without any guessing. We note that ratio \alpha  -\epsilon can be achieved using  {(1/\epsilon)}^{O(1/\epsilon^4)}n\log n value oracle calls, but this algorithm is impractical already for large } values of \epsilon.
After revision:  Extending a result of Khuller, Moss, and Naor for the Budgeted Coverage problem, Sviridenko showed that the greedy algorithm combined with guessing 3 most profitable elements of an optimal solution has approximation ratio \alpha = 1-\frac{1 \approx 0.632 . We show that just 2 guesses sufficeto achieve ratio \alpha , we get an almost as good approximation ratio of 0.6174>0.9767\alpha in roughly O(n^2) time. Prior to our work, the only algorithms that were known to obtain an approximation ratio close to \alpha for BSM were the algorithm of Sviridenko and an algorithm of Ene -\epsilon can be achieved using {(1/\epsilon)}^{O(1/\epsilon^4)}n\log n value oracle calls, but this algorithm is impractical already for large } values of \epsilon.