Set Cover with Almost Consecutive Ones

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...