exactfive

Exactfive is a term used in combinatorial optimization and puzzle design to denote problems in which a fixed subset of five elements must be chosen from a finite universe to satisfy constraints or maximize a criterion. Instances arise in resource allocation, feature selection, scheduling, and game design, where the fixed cardinality of five shapes the search space and the interaction of constraints.

Formally, given a universe U and a collection of predicates or objectives {P_j} defined on subsets of U, an exactfive problem seeks a subset S ⊆ U with |S| = 5 such that all constraints P_j(S) hold, or that a scoring function f(S) is optimized subject to |S| = 5. Variants include exactfive coverage, where the five elements must together cover a set of requirements, and exactfive packing, where selected elements must avoid conflicts.

Origin and usage: The term has appeared in constraint programming discussions, SAT encodings, and puzzle communities as a straightforward cardinality constraint. It is used to study algorithmic performance for fixed-size subset selection and to illustrate reductions to and from related problem classes.

Algorithms and complexity: Exactfive is typically NP-hard in general, even with simple constraint structures. Solvers use backtracking, branch-and-bound, constraint programming, SAT/CP-SAT encodings, or integer programming. Heuristics and parameterized approaches exploit the fixed size to prune search efficiently.

See also: k-subset selection, cardinality constraints, exact cover, constraint satisfaction problems.