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{{Short description|Type of computational problem}}

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 [[Optimization (mathematics)|minimization problem]]s and usually [[integer linear programs]], whose [[dual problem]]s are called [[packing problems]].

Covering problems are usually [[dual problem|dual]] to [[packing problem]]s.

ImportantThe most prominent examples of covering problems are the [[Setset cover problem]], which is equivalent to the [[Hitting set|Hittinghitting Setset Problemproblem]], and its special cases, the [[Vertexvertex cover problem]] and the [[Edgeedge cover problem]]. The set cover problem is dual to the [[set packing|set packing problem]].

Covering problems allow the covering primitives to overlap; the process of covering something with non-overlapping primitives is called [[decomposition (disambiguation)|decomposition]].

{{Covering-Packing_Problem_Pairs}}

==General linear programming formulation==

==LP 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 (mathematics)|matrix]], the objective function, and right-hand side are nonnegative.<ref>{{harvtxtCite book |V. last=Vazirani |2001}}</ref> Letfirst=Vijay <math>\mathbf{b},\mathbf{c}</math> be vectors and <math>A</math> be a matrix, all with no negative entriesV. Then| oneauthor-link=Vijay canVazirani see| thetitle=Approximation followingAlgorithms [[integer| linearyear=2001 program]]| aspublisher=Springer-Verlag the| mostisbn=3-540-65367-8 general covering problem:}}{{rp|112}}

</ref> More precisely, consider the following general [[integer linear program]]:

{|

| minimize

| <math>\sum_{i=1}^n c_i x_i</math>

|-

| subject to

| <math> \sum_{i=1}^n a_{ji} x_i \geq b_j \text{ for }j=1,\dots,m</math>

|-

|

| <math>x_i \in \left\{0, 1, 2, \ldots\right\}\text{ for }i=1,\dots,n</math>.

|}

Such an integer linear program is called a '''covering problem''' if <math>a_{ji}, b_j, c_i \geq 0</math> for all <math>i=1,\dots,n</math> and <math>j=1,\dots,m</math>.

'''Intuition:''' Assume having <math>n</math> types of object and each object of type <math>i</math> has an associated cost of <math>c_i</math>. The number <math>x_i</math> indicates how many objects of type <math>i</math> we buy. If the constraints <math>A\mathbf{x}\geq \mathbf{b}</math> are satisfied, it is said that ''<math>\mathbf{x}</math> 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.<ref>{{cite web|author = Douek-Pinkovich, Y., Ben-Gal, I., & Raviv, T. (2022)|title = The Stochastic Test Collection Problem: Models, Exact and Heuristic Solution Approaches |url = http://www.eng.tau.ac.il/~bengal/STCP.pdf|publisher = European Journal of Operational Research, 299 (2022), 945–959} }}</ref>

[[File:Covering problem of Rado.gif|thumb|500px|The [[covering problem of Rado]], where a series of squares with parallel edges needs to cover an area of 1. For any set meeting these conditions, a subset of these squares is selected (indicated by the red coloring) in which no two squares overlap, and the total area is maximized. The goal is to make an arrangement of squares so that the total area of the optimal subset is ''minimized''. The examples each have maximal areas of 1/4, but there are some which have slightly lower.

<ref>{{citation

| last = Ajtai | first = Miklós | author-link = Miklós Ajtai

| journal = Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques

| mr = 319053

| pages = 61–63

| title = The solution of a problem of T. Radó

| volume = 21

| year = 1973}}</ref>

<ref>{{citation

| last1 = Bereg | first1 = Sergey

| last2 = Dumitrescu | first2 = Adrian

| last3 = Jiang | first3 = Minghui

| doi = 10.1007/s00453-009-9298-z

| issue = 3

| journal = Algorithmica

| mr = 2609053

| pages = 538–561

| title = On covering problems of Rado

| volume = 57

| year = 2010}}; preliminary announcement in [[SWAT and WADS conferences|SWAT 2008]], {{doi|10.1007/978-3-540-69903-3_27}}</ref>]]

[[File:DiscCoveringExample.svg|thumb|200px|The [[disk covering problem]], which asks what the smallest real number <math>r(n)</math> is such that <math>n</math> disks of radius <math>r(n)</math> can be arranged in such a way as to cover the unit disk.]]

=== Covering in Petri nets ===

For [[Petri net]]s, 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.

=== Rainbow covering{{Anchor|rainbow}} ===

In some covering problems, the covering should satisfy some additional requirements. In particular, in the '''rainbow covering''' problem, each of the original objects has a "color", and it is required that the covering contains exactly one (or at most one) object of each color. Rainbow covering was studied e.g. for covering points by [[interval (mathematics)|intervals]]:<ref>{{Cite journal|last1=Arkin|first1=Esther M.|last2=Banik|first2=Aritra|last3=Carmi|first3=Paz|last4=Citovsky|first4=Gui|last5=Katz|first5=Matthew J.|last6=Mitchell|first6=Joseph S. B.|last7=Simakov|first7=Marina|date=2018-12-11|title=Selecting and covering colored points|journal=Discrete Applied Mathematics|language=en|volume=250|pages=75–86|doi=10.1016/j.dam.2018.05.011|issn=0166-218X|doi-access=free}}</ref>

* There is a set ''J'' of ''n'' colored intervals on the [[real line]], and a set ''P'' of points on the real line.

* A [[subset]] ''Q'' of ''J'' is called a ''rainbow set'' if it contains at most a single interval of each color.

* A set of intervals ''J'' is called a ''covering'' of ''P'' if each point in ''P'' is contained in at least one interval of ''Q''.

* The ''Rainbow covering problem'' is the problem of finding a rainbow set ''Q'' that is a covering of ''P''.

The problem is [[NP-hardness|NP-hard]] (by reduction from [[linear SAT]]).

=== Conflict-free covering ===

: minimize <math>\mathbf{c}^T \mathbf{x}</math>

A more general notion is '''conflict-free covering'''.<ref>{{Cite journal|last1=Banik|first1=Aritra|last2=Sahlot|first2=Vibha|last3=Saurabh|first3=Saket|date=2020-08-01|title=Approximation algorithms for geometric conflict free covering problems|url=http://www.sciencedirect.com/science/article/pii/S0925772119301324|journal=Computational Geometry|language=en|volume=89|pages=101591|doi=10.1016/j.comgeo.2019.101591|s2cid=209959954 |issn=0925-7721|url-access=subscription}}</ref> In this problem:

: subject to <math>A \mathbf{x}\geq \mathbf{b}, \mathbf{x}\in\{0,1\}</math>

* There is a set ''O'' of ''m'' objects, and a conflict-graph ''G_O'' on ''O''.

The <math>i</math>th entry of the vector <math>\mathbf{x}</math> tells us, whether we take object <math>i</math> into our cover or not. The constraints <math>A\mathbf{x}\geq \mathbf{b}</math> tell us, what it means for <math>\mathbf{x}</math> to cover all object. Finally the cost function <math>\mathbf{c}</math> defines what we consider to be a 'cheap' cover.

* A subset ''Q'' of ''O'' is called ''conflict-free'' if it is an [[Independent set (graph theory)|independent set]] in ''G_O'', that is, no two objects in ''Q'' are connected by an edge in ''G_O''.

* A rainbow set is a conflict-free set in the special case in which ''G_O'' is made of disjoint cliques, where each clique represents a color.

''Conflict-free set cover'' is the problem of finding a conflict-free subset of ''O'' that is a covering of ''P''. Banik, Panolan, Raman, Sahlot and Saurabh<ref>{{Cite journal|last1=Banik|first1=Aritra|last2=Panolan|first2=Fahad|last3=Raman|first3=Venkatesh|last4=Sahlot|first4=Vibha|last5=Saurabh|first5=Saket|date=2020-01-01|title=Parameterized Complexity of Geometric Covering Problems Having Conflicts|url=https://doi.org/10.1007/s00453-019-00600-w|journal=Algorithmica|language=en|volume=82|issue=1|pages=1–19|doi=10.1007/s00453-019-00600-w|s2cid=254027914 |issn=1432-0541|url-access=subscription}}</ref> [[mathematical proof|prove]] the following for the special case in which the conflict-graph has bounded [[arboricity]]:

==Other uses==

For [[Petri net]]s, 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.

* If the geometric cover problem is [[Fixed-parameter algorithm|fixed-parameter]] tractable (FPT), then the conflict-free geometric cover problem is FPT.

==Notes==

* If the geometric cover problem admits an r-approximation algorithm, then the conflict-free geometric cover problem admits a similar approximation algorithm in FPT time.

{{reflist}}

==References==

{{reflist}}

* {{Cite book | last=V. Vazirani | first=Vijay | authorlink=Vijay Vazirani | coauthors= | title=Approximation Algorithms | year=2001 | publisher=Springer-Verlag | ___location= | isbn=3-540-65367-8 | pages=}}

[[Category:CombinatoricsCovering problems|*]]

[[Category:Theoretical computer science]]

[[Category:Mathematical problems]]