Set of primitive shapes whose union equals a polygon
In geometry, a covering of a polygon is a set of primitive units (e.g. squares) whose union equals the polygon. A polygon covering problem is a problem of finding a covering with a smallest number of units for a given polygon. This is an important class of problems in computational geometry. There are many different polygon covering problems, depending on the type of polygon being covered. An example polygon covering problem is: given a rectilinear polygon, find a smallest set of squares whose union equals the polygon.
In some scenarios, it is not required to cover the entire polygon but only its edges (this is called polygon edge covering) or its vertices (this is called polygon vertex covering).
In a covering problem, the units in the covering are allowed to overlap, as long as their union is exactly equal to the target polygon. This is in contrast to a packing problem, in which the units must be disjoint and their union may be smaller than the target polygon, and to a polygon partition problem, in which the units must be disjoint and their union must be equal to the target polygon.
A polygon covering problem is a special case of the set cover problem. In general, the problem of finding a smallest set covering is NP-complete, but for special classes of polygons, a smallest polygon covering can be found in polynomial time.
A covering of a polygon P is a collection of maximal units, possibly overlapping, whose union equals P.
A minimal covering is a covering that does not contain any other covering (i.e. it is a local minimum).
A minimum covering is a covering with a smallest number of units (i.e. a global minimum). Every minimum covering is minimal, but not vice versa.
In geometry, a covering of a polygon is a set of primitive units (e.g. squares) whose union equals the polygon. A polygoncovering problem is a problem...
the polygon (see Polygoncovering). It is possible to distinguish several types of squares/rectangles contained in a certain rectilinear polygon P: A...
decomposition is often used as a general term that includes both polygoncovering and partitioning. Polygon decomposition is applied in several areas: Pattern recognition...
In combinatorics and computer science, covering problems are computational problems that ask whether a certain combinatorial structure 'covers' another...
detailed explanation. Set covering is equivalent to the hitting set problem. That is seen by observing that an instance of set covering can be viewed as an...
program (ILP). This ILP belongs to the more general class of ILPs for covering problems. The integrality gap of this ILP is 2 {\displaystyle 2} , so its...
edge coverings in two graphs (the set C is marked with red). A minimum edge covering is an edge covering of smallest possible size. The edge covering number...
r={\epsilon \over 8M}} and m {\displaystyle m} is the number of samples. Polygoncovering Kissing number Shalev-Shwartz, Shai; Ben-David, Shai (2014). Understanding...
dominating set problem are also covering LPs. Finding a fractional coloring of a graph is another example of a covering LP. In this case, there is one...
packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely...
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the...
polygon. Polygon triangle covering, in which the triangles may overlap. Tiling by polygons, where the goal is to cover the entire plane with polygons...
bins, such that no remaining item fits into an unfilled bin. In the bin covering problem, the bin size is bounded from below: the goal is to maximize the...
event known as Kankaria Carnival. The reservoir is a 34-sided regular polygoncovering an area of 76 acres and having a shore length of approximately one...
ISBN 0-13-215509-5. Benchmarks with Hidden Optimum Solutions for Set Covering, Set Packing and Winner Determination Solving packaging problem in PHP...
In geometry, a digon, or a 2-gon, is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either...
thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees. The regular triacontagon is a constructible polygon, by an edge-bisection...
can form double coveringpolygons 2{p/q}, namely: t{7/6}={14/6}=2{7/3}, t{7/4}={14/4}=2{7/2}, and t{7/2}={14/2}=2{7}. An isotoxal polygon can be labeled...
In the bin covering problem, items of different sizes must be packed into a finite number of bins or containers, each of which must contain at least a...
used to address people present in an individual cell sector or large polygonscovering a complete city or country. Cell Broadcast messages can be updated...
form double coverings: t{9/8}={18/8}=2{9/4}, t{9/4}={18/4}=2{9/2}, t{9/2}={18/2}=2{9}. A regular skew octadecagon is the Petrie polygon for a number...
might not be under surveillance. Covering a rectilinear polygon with star polygons Star-shaped polygon, a class of polygon for which the art gallery problem...
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