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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 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 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.
Conversely, if C = {S v : v ∈ D} is a feasible solution of the set cover problem, then D is a dominating set for G, with | D | = | C |. Hence the size of a minimum dominating set for G equals the size of a minimum set cover for (U, S). Furthermore, there is a simple algorithm that maps a dominating set to a set cover of the same size and vice ...
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).
A set of points is said to guard a polygon if, for every point in the polygon, there is some such that the line segment between and does not leave the polygon. The art gallery problem can be applied in several domains such as in robotics , when artificial intelligences (AI) need to execute movements depending on their surroundings.
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.
A set of objects, some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly. Usually the packing must be without overlaps between goods and other goods or the container walls.