minimumsete

Minimumsete is a term used in mathematical and computational contexts to denote a subset of a given universe that satisfies a specified condition and is chosen according to a criterion of minimality. Because the phrase is not universally standardized, its exact meaning can vary by discipline and problem formulation. Two common variants appear in literature.

Inclusion-minimal minimumsete: A subset S of a universal set X satisfies a property P, and no proper subset of S also satisfies P. In this usage, minimumset refers to minimality with respect to inclusion: removing any element would violate the property.

Cardinality-minimal (minimum-cardinality) minimumsete: A subset S satisfies P and has the smallest possible size among all subsets of X that satisfy P. Here the emphasis is on achieving the fewest elements rather than on the structure of inclusion.

Examples help illustrate the distinction. If P(S) requires S to intersect each class in a partition of X, then any S that meets all classes with the smallest possible size may be a cardinality-minimal minimumsete; among those, some may be inclusion-minimal as well.

Applications of the concept include combinatorial optimization, logic simplification, database query minimization, and network design, where one seeks compact explanations, coverings, or representatives under a given constraint.

Relation to related notions: The term overlaps with minimal sets and minimum elements in order theory. It is related to problems such as set cover and hitting set, where one seeks either a smallest cover or a smallest hitting subset under given conditions.

See also: Minimal set, Set cover, Hitting set, Cardinality, Basis.

References: General discussions of minimal and minimum subsets in combinatorics and algorithmic problem definitions.