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In the fractional set cover problem, it is allowed to select fractions of sets, rather than entire sets. A fractional set cover is an assignment of a fraction (a number in [0,1]) to each set in , such that for each element x in the universe, the sum of fractions of sets that contain x is at least 1. The goal is to find a fractional set cover in ...
The cover is said to be an open cover if each of its members is an open set. That is, each is contained in , where is the topology on X). [1] A simple way to get a subcover is to omit the sets contained in another set in the cover. Consider specifically open covers.
The most prominent examples of covering problems are the set cover problem, which is equivalent to the hitting set problem, and its special cases, the vertex cover problem and the edge cover problem. Covering problems allow the covering primitives to overlap; the process of covering something with non-overlapping primitives is called decomposition.
The set of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals ( n − 1, n + 1) , where n takes all integer values in Z , cover R {\displaystyle \mathbb {R} } but there is no finite subcover.
A vertex-cover (aka hitting set or transversal) in H is set T ⊆ V such that, for all hyperedges e ∈ E, it holds that T ∩ e ≠ ∅. The vertex-cover number (aka transversal number) of a hypergraph H is the smallest size of a vertex cover in H. It is often denoted by τ(H). [1]: 466 For example, if H is this 3-uniform hypergraph:
The exact cover problem is NP-complete [3] and is one of Karp's 21 NP-complete problems. [4] It is NP-complete even when each subset in S contains exactly three elements; this restricted problem is known as exact cover by 3-sets, often abbreviated X3C. [3] Knuth's Algorithm X is an algorithm that finds all solutions to an exact cover problem.
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The maximum set packing need not cover every possible element. In the exact cover problem, every element of should be contained in exactly one of the subsets. Finding such an exact cover is an NP-complete problem, even in the special case in which the size of all sets is 3 (this special case is called exact 3 cover or X3C).