Covering problems

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 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.

General LP Formulation

In the context of linear programming, one can think of any 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, let us consider the following general integer linear program:

minimize	${\displaystyle \sum _{i=1}^{n}c_{i}x_{i}}$
subject to	${\displaystyle \sum _{i=1}^{n}a_{ij}x_{i}\geq b_{j}}$	for all ${\displaystyle j=1,\dots ,m}$
	${\displaystyle x_{i}\geq 0}$	for all ${\displaystyle i=1,\dots ,n}$ .

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

Intuition: Assume we have ${\displaystyle n}$  types of object and each object of type ${\displaystyle i}$  has an associated cost of ${\displaystyle c_{i}}$ . The number ${\displaystyle x_{i}}$  says how many objects of type ${\displaystyle i}$  we buy. If the constraints ${\displaystyle A\mathbf {x} \geq \mathbf {b} }$  are satisfied, we say 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.

Other uses

For Petri nets, for example, the covering problem is defined as the question if for a given marking, there exists a run of the net, such that some larger (or equal) marking can be reached. Larger means here that all components are at least as large as the ones of the given marking and at least one is properly larger.

Notes

1. Vazirani (2001, p. 112)

References

• Vazirani, Vijay V. (2001). Approximation Algorithms. Springer-Verlag. ISBN 3-540-65367-8.