Algorithm for finding smallest collection of components

Question

I\'m looking for an algorithm to solve the following problem. I have a number of subsets (1-n) of a given set (a-h). I want to find the smallest collection of subsets that will

Answer 1

This problem is known as Set Basis, and it is NP-complete (Larry J. Stockmeyer: The set basis problem is NP-complete. Technical Report RC-5431, IBM, 1975). Its formulation as a graph problem is Bipartite Dimension. Since it is very hard to solve in general, it might be useful to look if there are any helpful properties of your data (e.g., are the sets small? Is the solution small? Can all sets occur?)

I cannot think of an easy ILP formulation. Instead, you could try to reduce the problem to Clique Cover, which is better studied, using either the reduction from Kou&Wong or the one from Nor et al.. I have coauthered a paper discussing algorithms for Clique Cover, and written a Clique cover solver with both an exact solver and two heuristics.