In combinatorics and computer science, covering problems are computational problems that ask whether a certain combinatorial structure 'covers' another, or how large the structure has to be to do that. Covering problems are minimization problems and usually integer linear programs, whose dual problems are called packing problems.

The most prominent examples of covering problems are the set cover problem, which is equivalent to the hitting set problem, and its special cases, the vertex cover problem and the edge cover problem.

Covering Problems allows the covering primitives to overlap, If you want to cover something with primitives that don't overlap is called Decomposition_(disambiguation)

General linear programming formulation

In the context of linear programming, one can think of any minimization linear program as a covering problem if the coefficients in the constraint matrix, the objective function, and right-hand side are nonnegative.^[1] More precisely, consider the following general integer linear program:

minimize	${\displaystyle \sum _{i=1}^{n}c_{i}x_{i}}$
subject to	${\displaystyle \sum _{i=1}^{n}a_{ji}x_{i}\geq b_{j}{\text{ for }}j=1,\dots ,m}$
	${\displaystyle x_{i}\in \left\{0,1,2,\ldots \right\}{\text{ for }}i=1,\dots ,n}$.

Such an integer linear program is called a covering problem if ${\displaystyle a_{ji},b_{j},c_{i}\geq 0}$ for all ${\displaystyle i=1,\dots ,n}$ and ${\displaystyle j=1,\dots ,m}$.

Intuition: Assume having ${\displaystyle n}$ types of object and each object of type ${\displaystyle i}$ has an associated cost of ${\displaystyle c_{i}}$. The number ${\displaystyle x_{i}}$ indicates how many objects of type ${\displaystyle i}$ we buy. If the constraints ${\displaystyle A\mathbf {x} \geq \mathbf {b} }$ are satisfied, it is said that ${\displaystyle \mathbf {x} }$ is a covering (the structures that are covered depend on the combinatorial context). Finally, an optimal solution to the above integer linear program is a covering of minimal cost.

Kinds of covering problems

There are various kinds of covering problems in graph theory, computational geometry and more; see Category:Covering problems. Other stochastic related versions of the problem can be found.^[2]

References