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Rainbow Matching
In the mathematical discipline of graph theory, a rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Definition Given an edge-colored graph , a rainbow matching in is a set of pairwise non-adjacent edges, that is, no two edges share a common vertex, such that all the edges in the set have distinct colors. A maximum rainbow matching is a rainbow matching that contains the largest possible number of edges. History Rainbow matchings are of particular interest given their connection to transversals of Latin squares. Denote by the complete bipartite graph on vertices. Every proper -edge coloring of corresponds to a Latin square of order . A rainbow matching then corresponds to a transversal of the Latin square, meaning a selection of positions, one in each row and each column, containing distinct entries. This connection between transversals of Latin squares and rainbow matchings in has inspired additional interest in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conical Hull
Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102/ref>''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> of these vectors is a vector of the form : \alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n where \alpha_i are non-negative real numbers. The name derives from the fact that a conical sum of vectors defines a cone (possibly in a lower-dimensional subspace). Conical hull The set of all conical combinations for a given set ''S'' is called the conical hull of ''S'' and denoted ''cone''(''S'') or ''coni''(''S''). That is, :\operatorname (S)=\left\. By taking ''k'' = 0, it follows the zero vector (origin) belongs to all conical hulls (since the summation becomes an empty sum). The conical hull of a set ''S'' is a convex set. In fact, it is the intersection ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rainbow Covering
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. 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, the objective function, and right-hand side are nonnegative. More precisely, consider the following general integer linear program: Such an integer linear program is called a covering problem if a_, b_j, c_i \geq 0 for all i=1,\dots,n and j=1,\dots,m. Intuition: Assume having ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rainbow-independent Set
In graph theory, a rainbow-independent set (ISR) is an independent set in a graph, in which each vertex has a different color. Formally, let be a graph, and suppose vertex set is partitioned into subsets , called "colors". A set of vertices is called a rainbow-independent set if it satisfies both the following conditions: * It is an independent set – every two vertices in are not adjacent (there is no edge between them); * It is a rainbow set – contains at most a single vertex from each color . Other terms used in the literature are independent set of representatives, independent transversal, and independent system of representatives. As an example application, consider a faculty with departments, where some faculty members dislike each other. The dean wants to construct a committee with members, one member per department, but without any pair of members who dislike each other. This problem can be presented as finding an ISR in a graph in which the nodes are the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rainbow-colorable Hypergraph
In graph theory, the term bipartite hypergraph describes several related classes of hypergraphs, all of which are natural generalizations of a bipartite graph. Property B and 2-colorability The weakest definition of bipartiteness is also called 2-colorability. A hypergraph ''H'' = (''V'', ''E'') is called 2-colorable if its vertex set ''V'' can be partitioned into two sets, ''X'' and ''Y'', such that each hyperedge meets both ''X'' and ''Y''. Equivalently, the vertices of ''H'' can be 2-colored so that no hyperedge is monochromatic. Every bipartite graph ''G'' = (''X''+''Y'', ''E'') is 2-colorable: each edge contains exactly one vertex of ''X'' and one vertex of ''Y'', so e.g. ''X'' can be colored blue and ''Y'' can be colored yellow and no edge is monochromatic. The property of 2-colorability was first introduced by Felix Bernstein in the context of set families; therefore it is also called Property B. Exact 2-colorability A stronger definition of bipartiteness is: a hyperg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rainbow Coloring
In graph theory, a path in an edge-colored graph is said to be rainbow if no color repeats on it. A graph is said to be rainbow-connected (or rainbow colored) if there is a rainbow path between each pair of its vertices. If there is a rainbow shortest path between each pair of vertices, the graph is said to be strongly rainbow-connected (or strongly rainbow colored). Definitions and bounds The rainbow connection number of a graph G is the minimum number of colors needed to rainbow-connect G, and is denoted by \text(G). Similarly, the strong rainbow connection number of a graph G is the minimum number of colors needed to strongly rainbow-connect G, and is denoted by \text(G). Clearly, each strong rainbow coloring is also a rainbow coloring, while the converse is not true in general. It is easy to observe that to rainbow-connect any connected graph G, we need at least \text(G) colors, where \text(G) is the diameter of G (i.e. the length of the longest shortest path). On the ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Packing Problems
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life 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 cover every region of the container, where objects are allowed to overlap. In a bin packing problem, you are given: * A ''container'', usually a two- or three-dimensional convex region, possibly of infinite size. Multiple containers may be given depending on the problem. * A set of ''objects'', some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly. Usually the packin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a dete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David S
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". was, according to the Hebrew Bible, the Kings of Israel and Judah, third king of the Kingdom of Israel (united monarchy), United Kingdom of Israel. In the Books of Samuel, he is described as a young shepherd and Lyre, harpist who gains fame by slaying Goliath, a champion of the Philistines, in southern Canaan. David becomes a favourite of Saul, the first king of Israel; he also forges David and Jonathan, a notably close friendship with Jonathan (1 Samuel), Jonathan, a son of Saul. However, under the paranoia that David is seeking to usurp the throne, Saul attempts to kill David, forcing the latter to go into hiding and effectively operate as a fugitive for several years. After Saul and Jonathan are both killed in battle against the Philistin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Garey
Michael Randolph Garey (born November 19, 1945) is a computer science researcher, and co-author (with David S. Johnson) of '' Computers and Intractability: A Guide to the Theory of NP-completeness''. He and Johnson received the 1979 Frederick W. Lanchester Prize from the Operations Research Society of America for the book. Garey earned his PhD in computer science in 1970 from the University of Wisconsin–Madison. He was employed by AT&T Bell Laboratories in the Mathematical Sciences Research Center from 1970 until his retirement in 1999. For his last 11 years with the organization, he served as its director. His technical specialties included discrete algorithms and computational complexity, approximation algorithms, scheduling theory, and graph theory. From 1978 until 1981 he served as Editor-in-Chief of the ''Journal of the Association for Computing Machinery''. In 1995, Garey was inducted as a Fellow of the Association for Computing Machinery A fellow is a concep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
