Keywords and Synonyms
Hitting set
Problem Definition
The Set Cover problem has as input a set R of m items, a set C of n subsets of R and a weight function \( { w \colon C \rightarrow \mathbb{Q} } \). The task is to choose a subset \( { C^{\prime} \subseteq C } \) of minimum weight whose union contains all items of R.
The sets R and C can be represented by an \( { m \times n } \) binary matrix A that consists of a row for every item in R and a column for every subset of R in C, where an entry \( { a_{i,j} } \) is 1 iff the ith item in R is part of the jth subset in C. Therefore, the Set Cover problem can be formulated as follows.
Input: An \( { m \times n } \) binary matrix A and a weight function w on the columns of A.
Task: Select some columns of A with minimum weight such that the submatrix A′ of A that is induced by these columns has at least one 1 in every row.
While Set Cover is NP-hard in general [4], it can be solved in polynomial time on instances whose columns can be...
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Notes
- 1.
The C1P can be defined symmetrically for columns; this article focuses on rows. Set Cover on instances with the C1P can be solved in polynomial time, e. g., with a linear programming approach, because the corresponding coefficient matrices are totally unimodular (see [9]).
- 2.
The last row of C allows to distinguish the columns belonging to A from those belonging to B.
Recommended Reading
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Dom, M. (2008). Set Cover with Almost Consecutive Ones. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_368
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