How do I make my implementation of greedy set cover faster?

Question

I came up with the following implementation for the Greedy Set Cover after much discussion regarding my original question here. From the help I received, I encoded the probl

Answer 1

I use a trick when I implemented the famous greedy algorithm for set cover (no weights) in Matlab. It is possible that you could extend this trick to the weighted case somehow, using set cardinality / set weight instead of set cardinality. Moreover, if you use NumPy library, exporting Matlab code to Python should be very easy.

Here is the trick:

1. (optional) I sorted the sets in descending order with respect to the cardinality (i.e. number of elements they contain). I also stored their cardinalities.
2. I select a set S, in my implementation it is the largest (i.e. first set of the list), and I count how many uncovered elements it contains. Let's say that it contains n uncovered elements.
3. Since now I know there is a set S with n uncovered elements, I don't need to process all the sets with cardinality lower than n elements, because they cannot be better than S. So I just need to search for the optimal set among the sets with cardinality at least n; with my sorting, we can focus on them easily.