exact cover

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exact cover

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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 ${\displaystyle \{S(\alpha _{i},\beta _{i})\vert 1\leq i\leq m\}}$ is called an (eventual) exact cover if every (sufficiently large) positive integer occurs in exactly one ${\displaystyle S(\alpha _{i},\beta _{i})}$. If ${\displaystyle \{S(\alpha _{i},\beta _{i})\}_{i=1}^{m}}$ is an eventual exact cover, then ${\displaystyle \textstyle \sum _{i=1}^{m}{\alpha _{i}^{-1}}=1}$. […] In 1926, Beatty studied the case ${\displaystyle \beta =0}$ and published the following result as a problem: ${\displaystyle S(\alpha _{1},0)}$ and ${\displaystyle S(\alpha _{2},0)}$ form an exact cover if and only if ${\displaystyle \alpha _{1}}$ is irrational and ${\displaystyle \alpha _{1}^{2}+\alpha _{2}^{2}=1}$.

   • 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.

• 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.

• tiling (geometry)

• Partition of a set on Wikipedia.Wikipedia
• Knuth's Algorithm X on Wikipedia.Wikipedia