There is a set of items that need to be covered. The subsets of these items are available. If we ask for the minimum number of these subsets that cover the whole items, the problem is the well-known Set Cover problem. Let denote the solution to the Set Cover problem by the minimum solution. However, we have a solution where a subset of items covers all items, and each subset in this solution covers some specific items individually. Let call this solution a minimal solution. I need to find out how we can find the minimum solution from the minimal solution? What is the best approximation factor to do this?
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