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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 geometric set cover problem is the special case of the set cover problem in geometric settings. The input is a range space = (,) where is a universe of points in and is a family of subsets of called ranges, defined by the intersection of and geometric shapes such as disks and axis-parallel rectangles.
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.
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 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).
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The maximum coverage problem is a classical question in computer science, computational complexity theory, and operations research. It is a problem that is widely taught in approximation algorithms. As input you are given several sets and a number . The sets may have some elements in common.