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Polygon Covering
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''). 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. Related problems I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Graph
In graph theory, a perfect graph is a Graph (discrete mathematics), graph in which the Graph coloring, chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices. The perfect graphs include many important families of graphs and serve to unify results relating Graph coloring, colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time, despite their greater complexity for non-perfect graphs. In addition, several important minimax theorems in combinatorics, including Dilworth's theorem and Mirsky's theorem on partially ordered sets, Kőnig's theorem (gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 allow the covering primitives to overlap; the process of covering something with non-overlapping primitives is called decomposition. 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. More precisely, consider the following general integ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudotriangle
In Euclidean plane geometry, a pseudotriangle (''pseudo-triangle'') is the simply connected subset of the plane that lies between any three mutually tangent convex sets. A pseudotriangulation (''pseudo-triangulations'') is a partition of a region of the plane into pseudotriangles, and a pointed pseudotriangulation is a pseudotriangulation in which at each vertex the incident edges span an angle of less than π. Although the words "pseudotriangle" and "pseudotriangulation" have been used with various meanings in mathematics for much longer, the terms as used here were introduced in 1993 by Michel Pocchiola and Gert Vegter in connection with the computation of visibility relations and bitangents among convex obstacles in the plane. Pointed pseudotriangulations were first considered by Ileana Streinu (2000, 2005) as part of her solution to the carpenter's ruler problem, a proof that any simple polygonal path in the plane can be straightened out by a sequence of continuous motions. Ps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3SAT
3sat (, ''Dreisat'') is a free-to-air German-language public service television channel. It is a generalist channel with a cultural focus and is jointly operated by public broadcasters from Germany ( ZDF, ARD), Austria ( ORF) and Switzerland ( SRG SSR). The coordinating broadcaster is ZDF, at whose Mainz facility the broadcasting centre with studios for in-house productions is located. History 3sat was established to broadcast cultural programmes, originally by satellite. The network was founded as a cooperative network by Germany's ZDF, Austria's ORF, and Switzerland's SRG SSR (formerly SRG SSR idée suisse). 3sat began broadcasting on 1 December 1984, with its first programme being simulcasted on FS2, TV DRS and ZDF. ZDF leads the cooperative, though decisions are reached through consensus of the cooperative's partners. In 1990, DFF, television broadcaster of the German Democratic Republic became a cooperative member of 3sat, and a name change to 4sat was considere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique Cover
In graph theory, a clique cover or partition into cliques of a given undirected graph is a collection of cliques that cover the whole graph. A minimum clique cover is a clique cover that uses as few cliques as possible. The minimum for which a clique cover exists is called the clique cover number of the given graph. Relation to coloring A clique cover of a graph may be seen as a graph coloring of the complement graph of , the graph on the same vertex set that has edges between non-adjacent vertices of . Like clique covers, graph colorings are partitions of the set of vertices, but into subsets with no adjacencies ( independent sets) rather than cliques. A subset of vertices is a clique in if and only if it is an independent set in the complement of , so a partition of the vertices of is a clique cover of if and only if it is a coloring of the complement of . Computational complexity The clique cover problem in computational complexity theory is the algorithmic problem of find ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Graph
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them. Formal definition Formally, an intersection graph is an undirected graph formed from a family of sets : S_i, \,\,\, i = 0, 1, 2, \dots by creating one vertex for each set , and connecting two vertices and by an edge whenever the corresponding two sets have a nonempty intersection, that is, : E(G) = \. All graphs are intersection graphs Any undirected graph may be represented as an intersection graph. For each vertex of , form a set consisting of the edges incident to ; then two such sets have a nonempty intersection if and only if the corresponding vertices share an edge. Therefore, is the intersection graph of the sets . provide a construction that is more ef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existential Theory Of The Reals
In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form \exists X_1 \cdots \exists X_n \, F(X_1,\dots, X_n), where the variables X_i are interpreted as having real number values, and where F(X_1,\dots X_n) is a quantifier-free formula involving equalities and inequalities of real polynomials. A sentence of this form is true if it is possible to find values for all of the variables that, when substituted into formula F, make it become true.. The decision problem for the existential theory of the reals is the problem of finding an algorithm that decides, for each such sentence, whether it is true or false. Equivalently, it is the problem of testing whether a given semialgebraic set is non-empty. This decision problem is NP-hard and lies in PSPACE, giving it significantly lower complexity than Alfred Tarski's quantifier elimination procedure for deciding statements in the first-orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner Tree Problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined objective function. One well-known variant, which is often used synonymously with the term Steiner tree problem, is the Steiner tree problem in graphs. Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals (but may include additional vertices) and minimizes the total weight of its edges. Further well-known variants are the ''Euclidean Steiner tree problem'' and the '' rectilinear minimum Steiner tree problem''. The Steiner tree problem in graphs can be seen as a generalization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
APX-hard
In computational complexity theory, the class APX (an abbreviation of "approximable") is the set of NP optimization problems that allow polynomial-time approximation algorithms with approximation ratio bounded by a constant (or constant-factor approximation algorithms for short). In simple terms, problems in this class have efficient algorithms that can find an answer within some fixed multiplicative factor of the optimal answer. An approximation algorithm is called an f(n)-approximation algorithm for input size n if it can be proven that the solution that the algorithm finds is at most a multiplicative factor of f(n) times worse than the optimal solution. Here, f(n) is called the ''approximation ratio''. Problems in APX are those with algorithms for which the approximation ratio f(n) is a constant c. The approximation ratio is conventionally stated greater than 1. In the case of minimization problems, f(n) is the found solution's score divided by the optimum solution's score, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Polygon
In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons. The sum of external angles of a simple polygon is 2\pi. Every simple polygon with n sides can be polygon triangulation, triangulated by n-3 of its diagonals, and by the art gallery theorem its interior is visible from some \lfloor n/3\rfloor of its vertices. Simple polygons are commonly seen as the input to computational geometry problems, including point in polygon testing, area computation, the convex hull of a simple polygon, triangulation, and Euclidean shortest paths. Other constructions in geometry related to simple polygons include Schwarz–Christoffel mapping, used to find conformal maps involving simple polygons, polygonalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Of David
The Star of David (, , ) is a symbol generally recognized as representing both Jewish identity and Judaism. Its shape is that of a hexagram: the compound of two equilateral triangles. A derivation of the Seal of Solomon was used for decorative and mystical purposes by Kabbalah, Kabbalistic Jews and Muslims. The hexagram appears occasionally in Jewish contexts since antiquity as a decorative motif, such as a stone bearing a hexagram from the arch of the 3rd–4th century Khirbet Shura synagogue. A hexagram found in a religious context can be seen in a Leningrad Codex, manuscript of the Hebrew Bible from 11th-century Cairo. Its association as a distinctive symbol for the Jewish people and their religion dates to 17th-century Prague. In the 19th century, the symbol began to be widely used by the History of the Jews in Europe, Jewish communities of Eastern Europe, ultimately coming to represent Jewish identity or religious beliefs."The Flag and the Emblem" (MFA). It became repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
