exact cover

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English

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Noun

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exact cover (plural exact covers)

  1. (topology, combinatorics) Given a collection S of subsets of a set X, a subcollection S* of S such that each element of X is contained in exactly one subset in S*.
    • 2000, R. Tijdeman, “Exact covers of balanced sequences and Fraenkel's conjecture”, in F. Halter-Koch, Robert F. Tichy, editors, Algebraic Number Theory and Diophantine Analysis: Proceedings of the International Conference, Walter de Gruyter, page 468:
      A finite set is called an (eventual) exact cover if every (sufficiently large) positive integer occurs in exactly one . If is an eventual exact cover, then . [] In 1926, Beatty [Bea] studied the case and published the following result as a problem: and form an exact cover if and only if is irrational and .
    • 2011, R. Lu, S. Liu, J. Zhang, Searching for Doubly Self-orthogonal Latin Squares, Jimmy Lee (editor), Principles and Practice of Constraint Programming: 17th International Conference CP 2011, Proceedings, Springer, LNCS 6876, page 542,
      It is straightforward to use clique algorithms to construct a (partial) solution of a given combinatorial problem which is represented as a set system. If the solution of the combinatorial problem corresponds to the exact cover of the set system, a substantially more efficient algorithm can be utilized because of this property.

Usage notes

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  • An exact cover may be defined as a partition that is comprised only of sets that are members of S.
  • The general problem of finding an exact cover, S*, for a given S and X is known as the exact cover problem. It is NP-complete.
  • If an element of X belongs to a set T in S*, that element is said to be covered by T.

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See also

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Further reading

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