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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 discrete unit disc cover problem is a geometric version of the general set cover problem which is NP-hard. [2] Many approximation algorithms have been devised for these problems. Due to the geometric nature, the approximation ratios for these problems can be much better than the general set cover/hitting set problems.
Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every planar set of diameter one. The diameter of a set by definition is the least upper bound of the distances between all pairs of points in the set. A shape covers a set if it contains a congruent subset.
Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. In a bin packing problem, people are given:
In theoretical computer science, monotone dualization is a computational problem of constructing the dual of a monotone Boolean function.Equivalent problems can also be formulated as constructing the transversal hypergraph of a given hypergraph, of listing all minimal hitting sets of a family of sets, or of listing all minimal set covers of a family of sets.
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 Book Loft covers 7,500 square feet of space, and along with books the store sells jigsaw puzzles, posters, and other merchandise. [7] Eighteen music systems each play different music to create a genre-specific soundtrack in each area. [8] The children's areas are the most popular rooms, according to owner Carl Jacobsma. [9]