Content deleted Content added
m Open access bot: doi added to citation with #oabot. |
Joel Brennan (talk | contribs) m added wikilinks |
||
Line 1:
{{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 [[linear programs]], whose [[dual problem]]s are called [[packing
The most prominent examples of covering problems are the [[set cover problem]], which is equivalent to the [[Hitting set|hitting set problem]], and its special cases, the [[vertex cover problem]] and the [[edge cover problem]].
Line 6:
{{Covering-Packing_Problem_Pairs}}
==General linear programming 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 (mathematics)|matrix]], the objective function, and right-hand side are nonnegative.<ref>{{Cite book | last=Vazirani | first=Vijay V. | author-link=Vijay Vazirani | title=Approximation Algorithms | year=2001 | publisher=Springer-Verlag | isbn=3-540-65367-8 }}{{rp|112}}
</ref> More precisely, consider the following general [[integer linear program]]:
{|
Line 23:
==Kinds of covering problems==
There are various kinds of covering problems in [[graph theory]], [[computational geometry]] and more
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.
== {{Anchor|rainbow}}Rainbow covering and conflict-free covering ==
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|last=Arkin|first=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|url=http://www.sciencedirect.com/science/article/pii/S0166218X18302695|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''.
Line 43:
* A rainbow set is a conflict-free set in the special case in which ''G<sub>O</sub>'' 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|last=Banik|first=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|issn=1432-0541}}</ref> [[mathematical proof|prove]] the following for the special case in which the conflict-graph has bounded [[arboricity]]:
* If the geometric cover problem is [[Fixed-parameter algorithm|fixed-parameter]] tractable (FPT), then the conflict-free geometric cover problem is FPT.
| |||