The set cover problem is a classical question in combinatorics, computer science, operations research, and complexity theory.
Given a set of elements {1, 2, …, n} (called the universe) and a collection S of m subsets whose union equals the universe, the set cover problem is to identify the smallest sub-collection of S whose union equals the universe. For example, consider the universe U = {1, 2, 3, 4, 5} and the collection of sets S = { {1, 2, 3}, {2, 4}, {3, 4}, {4, 5} }. Clearly the union of S is U. However, we can cover all elements with only two sets: { {1, 2, 3}, {4, 5} }, see picture. Therefore, the solution to the set cover problem has size 2.
More formally, given a universe and a family of subsets of , a set cover is a subfamily of sets whose union is .
The decision version of set covering is NP-complete. It is one of Karp's 21 NP-complete problems shown to be NP-complete in 1972. The optimization/search version of set cover is NP-hard.[1] It is a problem "whose study has led to the development of fundamental techniques for the entire field" of approximation algorithms.[2]
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